∫ ( cosn-1x/ sinn+1x)dx, (where n ≠ 0 is equal to

Question

∫ ( cosn-1x/ sinn+1x)dx, (where n ≠ 0 is equal to (A)  (cotn x/n) +C (B)  (-cotn-1 x/n-1)+C (C)  (-cotn x/n)+ C (D)  (cotn-1 x/n-1)+C

Tardigrade Admin · Accepted Answer

Let I =∫ ( cos n-1 x/ sin n+1 x) d x1 (n ≠ 0) =∫ ( cos n-1 x/ sin n+1 x) × ( sin 2 x/ sin 2 x) d x =∫ ( cos n-1 x/ sin n-1 x) ⋅ cosec 2 x d x =∫ cot n-1 x ⋅ cosec 2 x d x Let t = cot x ⇒ d t =- cosec 2 x d x=∫ tn-1(-d t) =-((tn/n))+C =-(1/n) cot n x+C (∵ t= cot x)

Q. $\int \frac{c o s ^{n - 1} x}{s i n ^{n + 1} x} d x, (w h ere n \neq = 0$ is equal to

$\frac{co t ^{n} x}{n} + C$

$\frac{- co t ^{n - 1} x}{n - 1} + C$

$\frac{- co t ^{n} x}{n} + C$

$\frac{co t ^{n - 1} x}{n - 1} + C$

Q. ∫sinn+1xcosn−1x​dx,(wheren=0 is equal to

ncotnx​+C

n−1−cotn−1x​+C

n−cotnx​+C

n−1cotn−1x​+C

Let I=∫sinn+1xcosn−1x​dx1​(n=0) =∫sinn+1xcosn−1x​×sin2xsin2x​dx =∫sinn−1xcosn−1x​⋅cosec2xdx =∫cotn−1x⋅cosec2xdx Let t=cotx ⇒dt=−cosec2xdx=∫tn−1(−dt) =−(ntn​)+C =−n1​cotnx+C(∵t=cotx)

Q. $\int \frac{c o s ^{n - 1} x}{s i n ^{n + 1} x} d x, (w h ere n \neq = 0$ is equal to

$\frac{co t ^{n} x}{n} + C$

$\frac{- co t ^{n - 1} x}{n - 1} + C$

$\frac{- co t ^{n} x}{n} + C$

$\frac{co t ^{n - 1} x}{n - 1} + C$