text: # Thread: Integration with change of variable
  
  1. ## Integration with change of variable
  
  Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?
  
  $\displaystyle \int cos(x)(1+sin^2(x)) dx$
  
  Thanks again!
  
  2. Originally Posted by aceband
  Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?
  
  $\displaystyle \int cos(x)(1+sin^2(x)) dx$
  
  Thanks again!
  Substitute $\displaystyle u = \sin x$ to get $\displaystyle \int 1 + u^2 \, du$ ....
  
  3. Originally Posted by aceband
  Hi i have this question but see what to do. My best guess is that u = 1 + sin^2x and perhaps that is an identity im not quite seeing?
  
  $\displaystyle \int cos(x)(1+sin^2(x)) dx$
  
  Thanks again!
  You're on the right track.
  
  Note that if you let $\displaystyle u = \sin{x}$ then $\displaystyle \frac{du}{dx} = \cos{x}$.
  
  So the integral becomes
  
  $\displaystyle \int{\cos{x}(1 + \sin^2{x})\,dx} = \int{(1 + u^2)\,\frac{du}{dx}\,dx}$
  
  $\displaystyle = \int{1 + u^2\,du}$
  
  $\displaystyle = u + \frac{1}{3}u^3 + C$
  
  $\displaystyle = \sin{x} + \frac{1}{3}\sin^3{x} + C$.