What is the integral of (cos(x))/(sin^{18)(x)} ?

The integral of (cos(x))/(sin^{18)(x)} is -1/17 csc^{17}(x)+C

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integral of (cos(x))/(sin^{18)(x)}

\int \frac{cos ( x )}{sin ^{18} ( x )} d x

- \frac{1}{17} csc^{17} (x) + C

Solution steps

\int \frac{cos ( x )}{sin ^{18} ( x )} d x

Apply u-substitution

: \int \frac{1}{u ^{18}} d u

= \int \frac{1}{u ^{18}} d u

Apply the Power Rule

: - \frac{1}{17 u ^{17}}

= - \frac{1}{17 u ^{17}}

Substitute back

u = sin (x)

= - \frac{1}{17 sin ^{17} ( x )}

Simplify

- \frac{1}{17 sin ^{17} ( x )} : - \frac{1}{17} csc^{17} (x)

= - \frac{1}{17} csc^{17} (x)

Add a constant to the solution

= - \frac{1}{17} csc^{17} (x) + C

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What is the integral of (cos(x))/(sin^{18)(x)} ?
The integral of (cos(x))/(sin^{18)(x)} is -1/17 csc^{17}(x)+C