Question

(atanx + bcotx)²,Integrate the given function w.r.t. x considering them well defined and integrable over proper domain.

Answer 1

In the attachment I have answered this problem.     The integrand is modified in such a way that it is suitable for integration.      See the attachment for detailed solution.

Answer 2

we have to find the value of $\int{(atanx+bcotx)^2}\,dx$


first of all, we have to solve (atanx + bcotx)²
(atanx + bcotx)² = a²tan²x + b²cot²x + 2abtanx.cotx
= a²(sec²x - 1) + b²(cosec² - 1) + 2ab
= a²sec²x + b²cosec²x - (a² + b² - 2ab)
= a²sec²x + b²cosec²x - (a - b)²

now, $\int{a^2sec^2x+b^2cosec^2x-(a-b)^2}\,dx$

$=a^2\int{sec^2x}\,dx+b^2\int{cosec^2x}\,dx-(a-b)^2\int{dx}$

$=a^2tanx+b^2(-cotx)-(a-b)^2x+C$

$=a^2tanx-b^2cotx-(a-b)^2x$