What is the integral of (-cos(x))/(sin^5(x)) ?

The integral of (-cos(x))/(sin^5(x)) is 1/4 csc^4(x)+C

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integral of (-cos(x))/(sin^5(x))

\int \frac{- cos ( x )}{sin ^{5} ( x )} d x

\frac{1}{4} csc^{4} (x) + C

Solution steps

\int \frac{- cos ( x )}{sin ^{5} ( 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 ^{5} ( x )} d x

Apply u-substitution

: \int \frac{1}{u ^{5}} d u

= - \int \frac{1}{u ^{5}} d u

Apply the Power Rule

: - \frac{1}{4 u ^{4}}

= - (- \frac{1}{4 u ^{4}})

Substitute back

u = sin (x)

= - (- \frac{1}{4 sin ^{4} ( x )})

Simplify

- (- \frac{1}{4 sin ^{4} ( x )}) : \frac{1}{4} csc^{4} (x)

= \frac{1}{4} csc^{4} (x)

Add a constant to the solution

= \frac{1}{4} csc^{4} (x) + C

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What is the integral of (-cos(x))/(sin^5(x)) ?
The integral of (-cos(x))/(sin^5(x)) is 1/4 csc^4(x)+C