Question

integrate cot ( 5x - 4 ) dx​

Answer 1

Answer:

$I=∫ \: \cot {}^{5}x \: cosec {}^{4}x \: dx$

$I=∫cot {}^{5}x \: cosec {}^{2} x \: cosec {}^{2}x \: dx$

$I=∫cot {}^{5}x \: (1 + cot {}^{2}x) \: cosec {}^{2}x \: dx$

$I = ∫cot {}^{5}x \: cosec {}^{2}x \: dx \: 2 ∫ \: cot {}^{7}x \: cosec {}^{2}x \: dx$

Let

$t=cotx⇒dt=−cosec {}^{2}xdx$

$I=−∫ \: t {}^{5} dt \: - \: ∫ t {}^{7} dt$

$= \: \frac{1}{6}t \: - \: \frac{1}{8}t {}^{8} + c$

$= - \frac{1}{6}cot {}^{6}x - \frac{1}{8}cot {}^{8}x + c \: where \: t = cot \: x$