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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \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

\frac{d}{d x} (3 x^{2} + 5 x + 1)

\int e^{2 x + 3} \cdot sin (4 x) d x

\int x^{2} (x^{3} + 5)^{7} d x

\frac{d}{d x} (x^{12} - 12 e^{x})

f (x) = e^{- 1}

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