Método de integración por cambio de variable

Este método, también llamado de sustitución, consiste en transformar la integral en otra más sencilla, usando un cambio de variable. Si queremos calcular una integral de la forma ∫ f(g(x)) g'(x) dx MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaacaWGMbGaaiikaiaadEgacaGGOaGaamiEaiaacMcacaGGPaGaam4zaiaacEcacaGGOaGaamiEaiaacMcacaWGKbGaamiEaaWcbeqab0Gaey4kIipaaaa@4342@ podemos introducir la nueva variable t = g(x), de modo que la diferencial dt=g'(x)dx MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadshacqGH9aqpcaWGNbGaai4jaiaacIcacaWG4bGaaiykaiaadsgacaWG4baaaa@3EA9@  y la integral se transforma en ∫ f(t)dt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaacaWGMbGaaiikaiaadshacaGGPaGaamizaiaadshaaSqabeqaniabgUIiYdaaaa@3D08@ . Una vez resuelta esta nueva integral se reemplaza en el resultado t = g(x) y se vuelve a la variable x.

Las fórmulas de cambio de variable para las integrales inmediatas son las que aparecen en la cuarta columna de la tabla de integrales.

La fórmula de cambio de variable se deduce de la regla de la cadena o derivada de una función compuesta.

E J E M P L O S   I= ∫ 1 x 1− ln⁡ 2 x dx MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9maapeaabaWaaSaaaeaacaaIXaaabaGaamiEamaakaaabaGaaGymaiabgkHiTiGacYgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaamiEaaWcbeaaaaGccaWGKbGaamiEaaWcbeqab0Gaey4kIipaaaa@430C@ ,   t=ln⁡x MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2da9iGacYgacaGGUbGaamiEaaaa@3ACE@ ,   dt= 1 x dx MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadshacqGH9aqpdaWcaaqaaiaaigdaaeaacaWG4baaaiaadsgacaWG4baaaa@3C84@   I= ∫ 1 1− t 2 dt =arcsin⁡t+K=arcsin⁡(ln⁡x)+K MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9maapeaabaWaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIXaGaeyOeI0IaamiDamaaCaaaleqabaGaaGOmaaaaaeqaaaaakiaadsgacaWG0baaleqabeqdcqGHRiI8aOGaeyypa0JaciyyaiaackhacaGGJbGaai4CaiaacMgacaGGUbGaamiDaiabgUcaRiaadUeacqGH9aqpciGGHbGaaiOCaiaacogacaGGZbGaaiyAaiaac6gaciGGOaGaciiBaiaac6gacaWG4bGaaiykaiabgUcaRiaadUeaaaa@55F1@   I= ∫ tan⁡x dx= ∫ sin⁡x cos⁡x dx MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9maapeaabaGaciiDaiaacggacaGGUbGaamiEaGGaaiab=bcaGiaadsgacaWG4bGaeyypa0daleqabeqdcqGHRiI8aOWaa8qaaeaadaWcaaqaaiGacohacaGGPbGaaiOBaiaadIhaaeaaciGGJbGaai4BaiaacohacaWG4baaaaWcbeqab0Gaey4kIipakiab=bcaGiaadsgacaWG4baaaa@4DB5@ ,   t=cos⁡x MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2da9iGacogacaGGVbGaai4CaiaadIhaaaa@3BBD@ ,   dt=−sin⁡x dx MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadshacqGH9aqpcqGHsislciGGZbGaaiyAaiaac6gacaWG4baccaGae8hiaaIaamizaiaadIhaaaa@404D@   I=− ∫ 1 t dt=−ln⁡t+K=−ln⁡(cos⁡x)+K MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiabg2da9iabgkHiTmaapeaabaWaaSaaaeaacaaIXaaabaGaamiDaaaaaSqabeqaniabgUIiYdaccaGccqWFGaaicaWGKbGaamiDaiabg2da9iabgkHiTiGacYgacaGGUbGaamiDaiabgUcaRiaadUeacqGH9aqpcqGHsislciGGSbGaaiOBaiaacIcaciGGJbGaai4BaiaacohacaWG4bGaaiykaiabgUcaRiaadUeaaaa@505D@

---