What is the integral of-(cos(x))/(sin^4(x)) ?

The integral of-(cos(x))/(sin^4(x)) is 1/3 csc^3(x)+C

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integral of-(cos(x))/(sin^4(x))

\int - \frac{cos ( x )}{sin ^{4} ( x )} d x

\frac{1}{3} csc^{3} (x) + C

Solution steps

\int - \frac{cos ( x )}{sin ^{4} ( 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 ^{4} ( x )} d x

Apply u-substitution

: \int \frac{1}{u ^{4}} d u

= - \int \frac{1}{u ^{4}} d u

Apply the Power Rule

: - \frac{1}{3 u ^{3}}

= - (- \frac{1}{3 u ^{3}})

Substitute back

u = sin (x)

= - (- \frac{1}{3 sin ^{3} ( x )})

Simplify

- (- \frac{1}{3 sin ^{3} ( x )}) : \frac{1}{3} csc^{3} (x)

= \frac{1}{3} csc^{3} (x)

Add a constant to the solution

= \frac{1}{3} csc^{3} (x) + C

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What is the integral of-(cos(x))/(sin^4(x)) ?
The integral of-(cos(x))/(sin^4(x)) is 1/3 csc^3(x)+C