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

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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

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

Apply the Power Rule

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

= - \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

y = \frac{7}{x} (1.7)

(\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}

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