# Integration with change of variable

• Dec 14th 2009, 04:00 AM
aceband
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!
• Dec 14th 2009, 04:02 AM
mr fantastic
Quote:

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 $\displaystyle \int 1 + u^2 \, du$ ....
• Dec 14th 2009, 04:05 AM
Prove It
Quote:

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

$\displaystyle \int{\cos{x}(1 + \sin^2{x})\,dx} = \int{(1 + u^2)\,\frac{du}{dx}\,dx}$

$\displaystyle = \int{1 + u^2\,du}$

$\displaystyle = u + \frac{1}{3}u^3 + C$

$\displaystyle = \sin{x} + \frac{1}{3}\sin^3{x} + C$.