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!

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- Dec 14th 2009, 04:00 AMacebandIntegration 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 AMmr fantastic
- Dec 14th 2009, 04:05 AMProve It
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

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

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

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

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