Propiedades de la integral indefinida

Propiedades básicas a tener en cuenta para calcular primitivas son

∫ ( f(x)+g(x) )dx = ∫ f(x)dx+ ∫ g(x)dx

∫ af(x)dx=a ∫ f(x)dx
para cualquier número a.

Si ∫ f(x)dx=F(x) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaacaWGMbGaaiikaiaadIhacaGGPaGaamizaiaadIhacqGH9aqpcaWGgbGaaiikaiaadIhacaGGPaaaleqabeqdcqGHRiI8aaaa@4137@ , entonces ∫ f(ax+b)dx= 1 a F(ax+b) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaacaWGMbGaaiikaiaadggacaWG4bGaey4kaSIaamOyaiaacMcacaWGKbGaamiEaiabg2da9maalaaabaGaaGymaaqaaiaadggaaaGaamOraiaacIcacaWGHbGaamiEaiabgUcaRiaadkgacaGGPaaaleqabeqdcqGHRiI8aaaa@4846@

Cuidado: la integral del producto de dos funciones no es el producto de las integrales.

Esas propiedades permiten descomponer una integral en otras más sencillas.

EJEMPLOS

∫ ( x 2 +1 x + x 5 ) dx= ∫ ( x+ 1 x + x 5/2 ) dx MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaadaqadaqaamaalaaabaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdaaeaacaWG4baaaiabgUcaRmaakaaabaGaamiEamaaCaaaleqabaGaaGynaaaaaeqaaaGccaGLOaGaayzkaaaaleqabeqdcqGHRiI8aGGaaOGae8hiaaIaamizaiaadIhacqGH9aqpdaWdbaqaamaabmaabaGaamiEaiabgUcaRmaalaaabaGaaGymaaqaaiaadIhaaaGaey4kaSIaamiEamaaCaaaleqabaGaaGynaiaac+cacaaIYaaaaaGccaGLOaGaayzkaaaaleqabeqdcqGHRiI8aOGae8hiaaIaamizaiaadIhaaaa@52D9@ = x 2 2 +ln⁡x+ 2 7 x 7/2 +K MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0ZaaSaaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaaaacqGHRaWkciGGSbGaaiOBaiaadIhacqGHRaWkdaWcaaqaaiaaikdaaeaacaaI3aaaaiaadIhadaahaaWcbeqaaiaaiEdacaGGVaGaaGOmaaaakiabgUcaRiaadUeaaaa@44F8@  

∫ [ sin⁡(3x+2)+5cos⁡(6x) ]dx=− 1 3 cos⁡(3x+2)+ 5 6 sin⁡(6x)+K MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaadaWadaqaaiGacohacaGGPbGaaiOBaiGacIcacaaIZaGaamiEaiabgUcaRiaaikdacaGGPaGaey4kaSIaaGynaiGacogacaGGVbGaai4CaiaacIcacaaI2aGaamiEaiaacMcaaiaawUfacaGLDbaacaWGKbGaamiEaiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaaiodaaaGaci4yaiaac+gacaGGZbGaaiikaiaaiodacaWG4bGaey4kaSIaaGOmaiaacMcacqGHRaWkdaWcaaqaaiaaiwdaaeaacaaI2aaaaiGacohacaGGPbGaaiOBaiGacIcacaaI2aGaamiEaiaacMcacqGHRaWkcaWGlbaaleqabeqdcqGHRiI8aaaa@5FE8@  

Comprobar el cálculo: Como la derivada de la primitiva F(x)= ∫ f(x)dx MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacIcacaWG4bGaaiykaiabg2da9maapeaabaGaamOzaiaacIcacaWG4bGaaiykaiaadsgacaWG4baaleqabeqdcqGHRiI8aaaa@4137@   debe ser la función original f(x) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@392F@ , siempre podemos comprobar que el resultado es correcto derivando F(x) MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacIcacaWG4bGaaiykaaaa@390F@