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

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

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

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

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

Solution steps

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

Simplify

= \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

\int (x + 2)^{4} d x

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

\int e^{- x^{3}} (- 3 x^{2}) d x

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