What is the integral of (cos(x))/(sin^8(x)) ?

The integral of (cos(x))/(sin^8(x)) is -1/7 csc^7(x)+C

integral of (cos(x))/(sin^8(x))

\int \frac{cos ( x )}{sin ^{8} ( x )} d x

- \frac{1}{7} csc^{7} (x) + C

Solution steps

\int \frac{cos ( x )}{sin ^{8} ( x )} d x

Apply u-substitution

= \int \frac{1}{u ^{8}} d u

Apply the Power Rule

= - \frac{1}{7 u ^{7}}

Substitute back

u = sin (x)

= - \frac{1}{7 sin ^{7} ( x )}

Simplify

- \frac{1}{7 sin ^{7} ( x )} : - \frac{1}{7} csc^{7} (x)

= - \frac{1}{7} csc^{7} (x)

Add a constant to the solution

= - \frac{1}{7} csc^{7} (x) + C

x^{2} d x + 2 y d y = 0, y (0) = 2

d er i v a t i v e (\frac{x}{3} + 9)^{\frac{3}{2}}

\int_{0}^{- 1} \frac{x ^{3} + 8}{x + 2} d x

\frac{d y}{d x} + \frac{y}{x} = x^{3} y^{2}

\frac{d}{d x} (ln e^{1 + x^{2}} + 3 x^{4} + cot^{2} (π x))

What is the integral of (cos(x))/(sin^8(x)) ?
The integral of (cos(x))/(sin^8(x)) is -1/7 csc^7(x)+C