Integration of x^2/(xsinx+cosx)^2

1. Mar 29, 2014

JasonHathaway

Hi everyone,

First of all, this isn't really a "homework", I've completed my calculus course and I'm just curious about this problem.

1. The problem statement, all variables and given/known data

$\int\frac{x^{2}}{(xsinx+cosx)^{2}} dx$

2. Relevant equations

Trigonometric substitutions, integration by parts maybe?

3. The attempt at a solution

This is a solved problem.

How does $\int\frac{x^{2}}{(xsinx+cosx)^{2}} dx$ become $\int xsecx \frac{xcosx}{(xsinx+cosx)^{2}} dx$?

2. Mar 29, 2014

jk22

Just because $$sec(x)=\frac{1}{cos(x)}$$

3. Mar 29, 2014

JasonHathaway

Did it multiply the numerator and denominator by $\frac{cosx}{cosx}$, which is $cosx secx$, and then both of $cosx$ and $secx$ took one "x" from the original numerator?

4. Mar 29, 2014

SammyS

Staff Emeritus
Yes.

In other words, $\ \cos(x)\cdot\sec(x) = 1 \ .$