228:@0.031627:0.974518:0.062330:0.974518:0.062330:0.956333:0.031627:0.956333:0.000000:0.000000:0.000000 Integración:@0.073089:0.043661:0.176062:0.043661:0.176062:0.024916:0.073089:0.024916:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 3.1 Integración por sustitución:@0.293660:0.299772:0.533825:0.299772:0.533825:0.282825:0.293660:0.282825:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Consideremos la integral con la antiderivada y la regla de la cadena:@0.293660:0.341280:0.779632:0.341280:0.779632:0.325567:0.293660:0.325567:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 #:@0.500926:0.383370:0.506043:0.383370:0.506043:0.365716:0.500926:0.365716:0.000000 '( ( )) :@0.505951:0.382787:0.560878:0.382787:0.560878:0.367075:0.505951:0.367075:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 F g x:@0.509909:0.382787:0.547312:0.382787:0.547312:0.367352:0.509909:0.367352:0.000000:0.000000:0.000000:0.000000:0.000000 ? :@0.560786:0.382233:0.571426:0.382233:0.571426:0.368517:0.560786:0.368517:0.000000:0.000000 g x dx F g x:@0.571334:0.382787:0.673236:0.382787:0.673236:0.367352:0.571334:0.367352:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 '( ):@0.580519:0.382787:0.600767:0.382787:0.600767:0.367075:0.580519:0.367075:0.000000:0.000000:0.000000:0.000000 5:@0.620518:0.382233:0.635851:0.382233:0.635851:0.368517:0.620518:0.368517:0.000000 ( ( )) :@0.646987:0.382787:0.686802:0.382787:0.686802:0.367075:0.646987:0.367075:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1 :@0.686710:0.382233:0.706553:0.382233:0.706553:0.368517:0.686710:0.368517:0.000000:0.000000 C.:@0.706461:0.382787:0.719622:0.382787:0.719622:0.367352:0.706461:0.367352:0.000000:0.000000 Por ejemplo, al evaluar :@0.293642:0.424295:0.466191:0.424295:0.466191:0.408582:0.293642:0.408582:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 #:@0.457227:0.424877:0.462344:0.424877:0.462344:0.407223:0.457227:0.407223:0.000000 e cosx dx e:@0.465621:0.424295:0.568131:0.424304:0.568131:0.408868:0.465621:0.408860:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 senx:@0.473288:0.418240:0.491199:0.418240:0.491199:0.409241:0.473288:0.409241:0.000000:0.000000:0.000000:0.000000 = a la :@0.546079:0.424304:0.599073:0.424304:0.599073:0.408591:0.546079:0.408591:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 senx:@0.598521:0.424304:0.629243:0.424304:0.629243:0.408868:0.598521:0.408868:0.000000:0.000000:0.000000:0.000000 + C consideraremos la regla de la cadena.:@0.629096:0.424304:0.926803:0.424304:0.926803:0.408591:0.629096:0.408591:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 O se sustituye una función cuya derivada está dentro de la integral::@0.293660:0.465811:0.778121:0.465811:0.778121:0.450099:0.293660:0.450099:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Hacemos :@0.293660:0.507319:0.366072:0.507319:0.366072:0.491606:0.293660:0.491606:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u:@0.458957:0.548826:0.468271:0.548826:0.468271:0.533391:0.458957:0.533391:0.000000 :@0.468198:0.548826:0.491428:0.548826:0.491428:0.533114:0.468198:0.533114:0.000000:0.000000:0.000000 5:@0.472155:0.548272:0.487489:0.548272:0.487489:0.534556:0.472155:0.534556:0.000000 senx:@0.491354:0.548826:0.522241:0.548826:0.522241:0.533391:0.491354:0.533391:0.000000:0.000000:0.000000:0.000000 así que :@0.522205:0.548826:0.581458:0.548826:0.581458:0.533114:0.522205:0.533114:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 du:@0.581447:0.541552:0.599818:0.541552:0.599818:0.526116:0.581447:0.526116:0.000000:0.000000 dx:@0.582276:0.557944:0.598990:0.557944:0.598990:0.542509:0.582276:0.542509:0.000000:0.000000 5 :@0.599855:0.548278:0.624208:0.548278:0.624208:0.534562:0.599855:0.534562:0.000000:0.000000:0.000000 cosx du cosx dx:@0.624116:0.548833:0.761551:0.548833:0.761551:0.533397:0.624116:0.533397:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 o :@0.657617:0.548833:0.671367:0.548833:0.671367:0.533120:0.657617:0.533120:0.000000:0.000000 5:@0.692959:0.548278:0.708292:0.548278:0.708292:0.534562:0.692959:0.534562:0.000000 #:@0.529171:0.590923:0.534288:0.590923:0.534288:0.573268:0.529171:0.573268:0.000000 :@0.534196:0.590340:0.538227:0.590340:0.538227:0.574627:0.534196:0.574627:0.000000 e du e :@0.538154:0.590340:0.605725:0.590345:0.605725:0.574910:0.538154:0.574905:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u:@0.545885:0.584281:0.551315:0.584281:0.551315:0.575282:0.545885:0.575282:0.000000 5:@0.576218:0.589790:0.591551:0.589790:0.591551:0.576075:0.576218:0.576075:0.000000 a la:@0.605670:0.590345:0.630096:0.590345:0.630096:0.574632:0.605670:0.574632:0.000000:0.000000:0.000000:0.000000 senx + C:@0.629986:0.590345:0.691319:0.590345:0.691319:0.574910:0.629986:0.574910:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ejemplo 1:@0.293648:0.631853:0.369431:0.631853:0.369431:0.615183:0.293648:0.615183:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Evaluemos:@0.293648:0.673360:0.376085:0.673360:0.376085:0.656691:0.293648:0.656691:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.375984:0.673360:0.411656:0.674530:0.411656:0.658817:0.375984:0.657647:0.000000:0.000000:0.000000:0.000000:0.000000 #:@0.379942:0.673943:0.385059:0.673943:0.385059:0.656288:0.379942:0.656288:0.000000 x:@0.411575:0.674530:0.419030:0.674530:0.419030:0.659095:0.411575:0.659095:0.000000 2:@0.418957:0.673975:0.434290:0.673975:0.434290:0.660260:0.418957:0.660260:0.000000 3:@0.434198:0.674530:0.443181:0.674530:0.443181:0.658817:0.434198:0.658817:0.000000 dx:@0.443107:0.674530:0.462987:0.674530:0.462987:0.659095:0.443107:0.659095:0.000000:0.000000:0.000000 3:@0.392832:0.664661:0.398069:0.664661:0.398069:0.655501:0.392832:0.655501:0.000000 .:@0.468439:0.673365:0.471642:0.673365:0.471642:0.657653:0.468439:0.657653:0.000000 Hacemos :@0.293662:0.714873:0.366074:0.714873:0.366074:0.699160:0.293662:0.699160:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u:@0.369721:0.756380:0.379035:0.756380:0.379035:0.740945:0.369721:0.740945:0.000000 5:@0.378961:0.756380:0.411100:0.756380:0.411100:0.740668:0.378961:0.740668:0.000000:0.000000:0.000000:0.000000 5:@0.382919:0.755825:0.398252:0.755825:0.398252:0.742110:0.382919:0.742110:0.000000 x :@0.411008:0.756380:0.440202:0.756380:0.440202:0.740945:0.411008:0.740945:0.000000:0.000000:0.000000:0.000000 2:@0.421647:0.755825:0.436981:0.755825:0.436981:0.742110:0.421647:0.742110:0.000000 3, entonces :@0.440128:0.756380:0.527592:0.756380:0.527592:0.740668:0.440128:0.740668:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 du :@0.527434:0.756380:0.549173:0.756380:0.549173:0.740945:0.527434:0.740945:0.000000:0.000000:0.000000 5 :@0.549118:0.755825:0.568961:0.755825:0.568961:0.742110:0.549118:0.742110:0.000000:0.000000 5:@0.568869:0.756380:0.577851:0.756380:0.577851:0.740668:0.568869:0.740668:0.000000 dx, :@0.577778:0.756380:0.600916:0.756380:0.600916:0.740945:0.577778:0.740945:0.000000:0.000000:0.000000:0.000000 es decir que :@0.600879:0.756380:0.732638:0.756380:0.732638:0.740668:0.600879:0.740668:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 dx:@0.692823:0.756380:0.709463:0.756380:0.709463:0.740945:0.692823:0.740945:0.000000:0.000000 5:@0.713366:0.755825:0.728699:0.755825:0.728699:0.742110:0.713366:0.742110:0.000000 du:@0.732683:0.750706:0.751054:0.750706:0.751054:0.735270:0.732683:0.735270:0.000000:0.000000 5:@0.737377:0.767098:0.746360:0.767098:0.746360:0.751385:0.737377:0.751385:0.000000 :@0.751090:0.755837:0.755692:0.755837:0.755692:0.742121:0.751090:0.742121:0.000000 y sustituimos:@0.755600:0.756392:0.850931:0.756392:0.850931:0.740679:0.755600:0.740679:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 #:@0.425411:0.798482:0.430528:0.798482:0.430528:0.780827:0.425411:0.780827:0.000000 :@0.430436:0.797899:0.434467:0.797899:0.434467:0.782186:0.430436:0.782186:0.000000 u:@0.448144:0.797897:0.457458:0.797897:0.457458:0.782461:0.448144:0.782461:0.000000 3:@0.438303:0.788028:0.443540:0.788028:0.443540:0.778867:0.438303:0.778867:0.000000 du:@0.461121:0.792217:0.479491:0.792217:0.479491:0.776782:0.461121:0.776782:0.000000:0.000000 5:@0.465815:0.808609:0.474798:0.808609:0.474798:0.792896:0.465815:0.792896:0.000000 5 :@0.479528:0.797348:0.517573:0.797348:0.517573:0.783632:0.479528:0.783632:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.503933:0.792217:0.512916:0.792217:0.512916:0.776504:0.503933:0.776504:0.000000 5:@0.503933:0.808609:0.512916:0.808609:0.512916:0.792896:0.503933:0.792896:0.000000 #:@0.517481:0.798485:0.522598:0.798485:0.522598:0.780831:0.517481:0.780831:0.000000 u du :@0.522506:0.797903:0.580224:0.797897:0.580224:0.782461:0.522506:0.782468:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.533026:0.788657:0.535486:0.788657:0.535486:0.778658:0.533026:0.778658:0.000000 3:@0.532616:0.798744:0.535885:0.798744:0.535885:0.788745:0.532616:0.788745:0.000000 5 :@0.561670:0.797342:0.593931:0.797348:0.593931:0.783632:0.561670:0.783626:0.000000:0.000000:0.000000 1:@0.580291:0.792217:0.589274:0.792217:0.589274:0.776504:0.580291:0.776504:0.000000 5:@0.580291:0.808609:0.589274:0.808609:0.589274:0.792896:0.580291:0.792896:0.000000 :@0.593839:0.796350:0.597870:0.796350:0.597870:0.780637:0.593839:0.780637:0.000000 u:@0.606289:0.790661:0.615603:0.790661:0.615603:0.775226:0.606289:0.775226:0.000000 4:@0.618585:0.778380:0.622322:0.778380:0.622322:0.768381:0.618585:0.768381:0.000000 3:@0.618819:0.788467:0.622087:0.788467:0.622087:0.778468:0.618819:0.778468:0.000000 4:@0.607584:0.806308:0.616567:0.806308:0.616567:0.790596:0.607584:0.790596:0.000000 3:@0.607584:0.822700:0.616567:0.822700:0.616567:0.806988:0.607584:0.806988:0.000000 1:@0.624022:0.797349:0.643865:0.797349:0.643865:0.783634:0.624022:0.783634:0.000000:0.000000 C =:@0.643773:0.797904:0.673077:0.797904:0.673077:0.782191:0.643773:0.782191:0.000000:0.000000:0.000000:0.000000 :@0.672967:0.797904:0.676280:0.797904:0.676280:0.782469:0.672967:0.782469:0.000000 3:@0.680850:0.792217:0.689833:0.792217:0.689833:0.776504:0.680850:0.776504:0.000000 20:@0.676359:0.808609:0.694324:0.808609:0.694324:0.792896:0.676359:0.792896:0.000000:0.000000 :@0.694379:0.797348:0.698981:0.797348:0.698981:0.783632:0.694379:0.783632:0.000000 (5:@0.698889:0.797903:0.712676:0.797903:0.712676:0.782190:0.698889:0.782190:0.000000:0.000000 x :@0.712584:0.797903:0.723279:0.797903:0.723279:0.782468:0.712584:0.782468:0.000000:0.000000 2:@0.723224:0.797348:0.738557:0.797348:0.738557:0.783632:0.723224:0.783632:0.000000 3):@0.738465:0.797903:0.752252:0.797903:0.752252:0.782190:0.738465:0.782190:0.000000:0.000000 4:@0.752801:0.784670:0.756537:0.784670:0.756537:0.774671:0.752801:0.774671:0.000000 3:@0.753035:0.794757:0.756303:0.794757:0.756303:0.784758:0.753035:0.784758:0.000000 :@0.757098:0.797897:0.760411:0.797897:0.760411:0.782461:0.757098:0.782461:0.000000 :@0.760346:0.791833:0.762278:0.791833:0.762278:0.782834:0.760346:0.782834:0.000000 1 :@0.762239:0.797342:0.782082:0.797342:0.782082:0.783626:0.762239:0.783626:0.000000:0.000000 C.:@0.781990:0.797897:0.795151:0.797897:0.795151:0.782461:0.781990:0.782461:0.000000:0.000000 Ejemplo 2:@0.293663:0.839404:0.369446:0.839404:0.369446:0.822734:0.293663:0.822734:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Evaluemos :@0.293663:0.880912:0.388737:0.880912:0.388737:0.865199:0.293663:0.865199:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 #:@0.379441:0.881494:0.384558:0.881494:0.384558:0.863840:0.379441:0.863840:0.000000 tanx dx.:@0.389860:0.880912:0.446287:0.880912:0.446287:0.865476:0.389860:0.865476:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Tengamos en cuenta que buscamos aquella función o aquellas :@0.446425:0.880912:0.931162:0.880912:0.931162:0.865199:0.446425:0.865199:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 funciones cuya derivada es :@0.293663:0.899190:0.493467:0.899190:0.493467:0.883477:0.293663:0.883477:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 tanx:@0.493217:0.899190:0.524582:0.899190:0.524582:0.883755:0.493217:0.883755:0.000000:0.000000:0.000000:0.000000 . Además, consideremos la definición de tangente:@0.524546:0.899190:0.886145:0.899190:0.886145:0.883477:0.524546:0.883477:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 #:@0.293660:0.942763:0.300638:0.942763:0.300638:0.918689:0.293660:0.918689:0.000000 senx:@0.304392:0.930767:0.334892:0.930767:0.334892:0.915332:0.304392:0.915332:0.000000:0.000000:0.000000:0.000000 cosx:@0.304373:0.947159:0.334892:0.947159:0.334892:0.931724:0.304373:0.931724:0.000000:0.000000:0.000000:0.000000 :@0.338666:0.940156:0.343268:0.940156:0.343268:0.926440:0.338666:0.926440:0.000000 dx :@0.343176:0.940711:0.363055:0.940711:0.363055:0.925275:0.343176:0.925275:0.000000:0.000000:0.000000 hacemos:@0.363000:0.940711:0.429653:0.940711:0.429653:0.924998:0.363000:0.924998:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 u cosx :@0.429524:0.940711:0.497289:0.940711:0.497289:0.925275:0.429524:0.925275:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.445262:0.940156:0.460596:0.940156:0.460596:0.926440:0.445262:0.926440:0.000000 asi que:@0.497244:0.940711:0.548582:0.940711:0.548582:0.924998:0.497244:0.924998:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 du senx dx :@0.548453:0.940711:0.661051:0.940711:0.661051:0.925275:0.548453:0.925275:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5 2:@0.573395:0.940156:0.607209:0.940156:0.607209:0.926440:0.573395:0.926440:0.000000:0.000000:0.000000 es decir que:@0.661032:0.940711:0.749184:0.940711:0.749184:0.924998:0.661032:0.924998:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 du senxdx.:@0.749019:0.940711:0.858089:0.940711:0.858089:0.925275:0.749019:0.925275:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.752258:0.940156:0.767592:0.940156:0.767592:0.926440:0.752258:0.926440:0.000000 5:@0.789183:0.940156:0.804517:0.940156:0.804517:0.926440:0.789183:0.926440:0.000000 Tema 3. Técnicas de integración :@0.073089:0.097630:0.545983:0.097630:0.545983:0.066388:0.073089:0.066388:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Evalúa la siguiente integral::@0.293660:0.171050:0.482842:0.171050:0.482842:0.156052:0.293660:0.156052:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 #:@0.577671:0.194640:0.582555:0.194640:0.582555:0.177788:0.577671:0.177788:0.000000 xdx.:@0.582555:0.194084:0.608525:0.194084:0.608525:0.179350:0.582555:0.179350:0.000000:0.000000:0.000000:0.000000 Pista: la integral de x se resuelve usando la regla básica de potencias.:@0.293660:0.217118:0.770262:0.217118:0.770262:0.202119:0.293660:0.202119:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Reto matemático:@0.302768:0.140818:0.437491:0.140818:0.437491:0.123871:0.302768:0.123871:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 EBC 11. 1. Comprende la relación entre la integral definida y el área de la región bajo una curva en el plano cartesiano.:@0.296794:0.971366:0.915191:0.971366:0.915191:0.959939:0.296794:0.959939:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.080328:0.224503:0.085739:0.224503:0.085739:0.215127:0.080328:0.215127:0.000000 b:@0.204832:0.224801:0.210967:0.224801:0.210967:0.214670:0.204832:0.214670:0.000000 y:@0.078272:0.152450:0.083105:0.152450:0.083105:0.143075:0.078272:0.143075:0.000000 x:@0.249410:0.224503:0.254122:0.224503:0.254122:0.215127:0.249410:0.215127:0.000000 f x:@0.228741:0.151636:0.240024:0.151636:0.240024:0.141505:0.228741:0.141505:0.000000:0.000000:0.000000 ( ):@0.231852:0.151636:0.243257:0.151636:0.243257:0.141323:0.231852:0.141323:0.000000:0.000000:0.000000 A :@0.116835:0.234135:0.125581:0.234135:0.125581:0.224004:0.116835:0.224004:0.000000:0.000000 5 :@0.125581:0.233771:0.150990:0.233771:0.150990:0.224768:0.125581:0.224768:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 f x dx:@0.150990:0.234135:0.176582:0.234135:0.176582:0.224004:0.150990:0.224004:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ( ):@0.154101:0.234135:0.165506:0.234135:0.165506:0.223822:0.154101:0.223822:0.000000:0.000000:0.000000 a:@0.109675:0.224805:0.115811:0.224805:0.115811:0.214673:0.109675:0.214673:0.000000 A:@0.173618:0.192946:0.180169:0.192946:0.180169:0.182814:0.173618:0.182814:0.000000 #:@0.141107:0.235475:0.145382:0.235475:0.145382:0.220727:0.141107:0.220727:0.000000 b:@0.146415:0.226922:0.150283:0.226922:0.150283:0.220475:0.146415:0.220475:0.000000 a:@0.146415:0.238507:0.150283:0.238507:0.150283:0.232060:0.146415:0.232060:0.000000 Imagina que estás :@0.082576:0.353984:0.206040:0.353984:0.206040:0.339700:0.082576:0.339700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 inflando un globo. La :@0.082576:0.369113:0.226111:0.369113:0.226111:0.354829:0.082576:0.354829:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 cantidad de aire que entra :@0.082576:0.384242:0.259764:0.384242:0.259764:0.369958:0.082576:0.369958:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 al globo depende de la :@0.082576:0.399371:0.239456:0.399371:0.239456:0.385087:0.082576:0.385087:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 presión que ejerces y de :@0.082576:0.414500:0.246104:0.414500:0.246104:0.400216:0.082576:0.400216:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 la elasticidad del globo. :@0.082576:0.429629:0.240773:0.429629:0.240773:0.415345:0.082576:0.415345:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Si queremos calcular la :@0.082576:0.444758:0.236988:0.444758:0.236988:0.430474:0.082576:0.430474:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 cantidad total de aire que :@0.082576:0.459887:0.255258:0.459887:0.255258:0.445603:0.082576:0.445603:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 hemos introducido en el :@0.082576:0.475016:0.248685:0.475016:0.248685:0.460732:0.082576:0.460732:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 globo hasta cierto punto, :@0.082576:0.490145:0.252968:0.490145:0.252968:0.475861:0.082576:0.475861:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 podemos utilizar una :@0.082576:0.505274:0.225834:0.505274:0.225834:0.490990:0.082576:0.490990:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 integral.:@0.082576:0.520403:0.135453:0.520403:0.135453:0.506119:0.082576:0.506119:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 La función que describe la :@0.082576:0.535532:0.259429:0.535532:0.259429:0.521247:0.082576:0.521247:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 cantidad de aire que entra :@0.082576:0.550661:0.259764:0.550661:0.259764:0.536376:0.082576:0.536376:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 en función de la presión :@0.082576:0.565790:0.245590:0.565790:0.245590:0.551505:0.082576:0.551505:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 podría ser bastante :@0.082576:0.580919:0.213036:0.580919:0.213036:0.566634:0.082576:0.566634:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 compleja. Sin embargo, si :@0.082576:0.596047:0.254704:0.596047:0.254704:0.581763:0.082576:0.581763:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 hacemos un cambio de :@0.082576:0.611176:0.242434:0.611176:0.242434:0.596892:0.082576:0.596892:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 variable (una sustitución), :@0.082576:0.626305:0.254192:0.626305:0.254192:0.612021:0.082576:0.612021:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 podemos simplificar la :@0.082576:0.641434:0.235122:0.641434:0.235122:0.627150:0.082576:0.627150:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 integral y encontrar una :@0.082576:0.656563:0.244263:0.656563:0.244263:0.642279:0.082576:0.642279:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 solución más fácilmente.:@0.082576:0.671692:0.245517:0.671692:0.245517:0.657408:0.082576:0.657408:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Interdisciplinariedad :@0.078581:0.289830:0.244955:0.289830:0.244955:0.272883:0.078581:0.272883:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Matemática :@0.078581:0.306472:0.172624:0.306472:0.172624:0.290759:0.078581:0.290759:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y vida cotidiana:@0.078581:0.323114:0.192154:0.323114:0.192154:0.307401:0.078581:0.307401:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 #:@0.293660:0.238760:0.296917:0.238760:0.296917:0.227525:0.293660:0.227525:0.000000 :@0.296917:0.238389:0.299482:0.238389:0.299482:0.228390:0.296917:0.228390:0.000000 xdx:@0.299482:0.238389:0.314675:0.238389:0.314675:0.228567:0.299482:0.228567:0.000000:0.000000:0.000000 :@0.314675:0.238389:0.329563:0.238389:0.329563:0.228390:0.314675:0.228390:0.000000:0.000000:0.000000 5 1:@0.317240:0.238036:0.347565:0.238039:0.347565:0.229311:0.317240:0.229308:0.000000:0.000000:0.000000 x:@0.329571:0.233153:0.334596:0.233153:0.334596:0.223154:0.329571:0.223154:0.000000 2:@0.334466:0.229294:0.337798:0.229294:0.337798:0.223465:0.334466:0.223465:0.000000 2:@0.330826:0.245761:0.336542:0.245761:0.336542:0.235762:0.330826:0.235762:0.000000 :@0.347565:0.238392:0.350130:0.238392:0.350130:0.228393:0.347565:0.228393:0.000000 C:@0.350130:0.238392:0.356432:0.238392:0.356432:0.228569:0.350130:0.228569:0.000000 MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000