# Integration problem (algebraic+trigonometric function)

Gold Member

## Homework Statement

find integral of the function- (x^2 + cos^2)(cosec^2) / (1+x^2)

2. The attempt at a solution
I noticed the denominator is the derivative of Arctan(x), I tried integrating by parts with various choices for 1st and second function but all of them end up being more complicated, I tried factoring in sec^2(x) which also didn't go a long way. I would appreciate some help to proceed further with this integral

Staff Emeritus
Homework Helper
I tried factoring in sec^2(x) which also didn't go a long way.
What does this mean?

Gold Member
What does this mean?
It means I tried multiplying and dividing by sec^2(x), but still couldn't integrate it

Staff Emeritus
Homework Helper
You're overthinking it. Multiply the numerator out and go from there.

Gold Member
You're overthinking it. Multiply the numerator out and go from there.
you mean separating numerator and denominator as two different functions and then integrate by parts? I already tried that

Staff Emeritus
Homework Helper
No, you just want to simplify the integrand first.

Gold Member
No, you just want to simplify the integrand first.
what did you mean by "multiply the numerator out?" multiply cosec^2 inside?

Staff Emeritus
Homework Helper
Distribute the factor of ##csc^2 x## into the sum.

Gold Member
Distribute the factor of ##csc^2 x##.
done already. ## cosec^2(x) x^2 + cot^2(x) ## in the numerator
then i separated both terms but i'd have to integrate both of them by parts which is really long

Staff Emeritus