Integration problem (algebraic+trigonometric function)

[Krushnaraj Pandya]

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

 

[vela]

I tried factoring in sec^2(x) which also didn't go a long way.

What does this mean?

 

[Krushnaraj Pandya]

What does this mean?

It means I tried multiplying and dividing by sec^2(x), but still couldn't integrate it

 

[vela]

You're overthinking it. Multiply the numerator out and go from there.

 

[Krushnaraj Pandya]

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

 

[vela]

No, you just want to simplify the integrand first.

 

[Krushnaraj Pandya]

No, you just want to simplify the integrand first.

what did you mean by "multiply the numerator out?" multiply cosec^2 inside?

 

[vela]

Distribute the factor of $csc^2 x$ into the sum.

 

[Krushnaraj Pandya]

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

 

[vela]

You haven't really simplified the integrand. To me, the obvious thing to try here is a trig identity. You have csc^2 and cot^2...a natural choice comes to mind. See what happens.

 

[Krushnaraj Pandya]

You haven't really simplified the integrand. To me, the obvious thing to try here is a trig identity. You have csc^2 and cot^2...a natural choice comes to mind. See what happens.

It was so easy! I really did overthink it. Thanks a lot though :)