Question

(2tanx - 3cotx)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.     I have applied  decomposition method to find the anti derivative of the given function.      See the attachment for detailed solution.

Answer 2

Hello,

Solution:

$\int {(2 \: tan \: x - 3 \: cot \: x)}^{2} \\ \\ 4 {tan}^{2} x + 9 \: {cot}^{2} x - 12 \: tan \: x \: cot \: x \\ \\ 4 {tan}^{2} x + 9 \: {cot}^{2} x - 12 \\ \\ \\ 4\int {tan}^{2} x \: dx+ 9\int \: {cot}^{2} x \: dx- \int12 \: dx \\ \\ {tan}^{2} x = {sec}^{2} x - 1 \\ \\ {cot}^{2}x = {cosec}^{2} x - 1 \\ \\ so \\ 4\int \: {sec}^{2} x \: dx - \int4 \: dx + 9\int \: {cosec}^{2} x \: - \int9 \: dx - \int12dx \\ \\ 4 \int\: {sec}^{2} x \: dx + 9\int \: {cosec}^{2} x \: - 25 \int 1\: dx \\ \\ = 4 \: tan \: x - 9 \: cot \: x - 25x + c$
Hope it helps you.