What is the integral of csc^6(x)cot(x) ?

The integral of csc^6(x)cot(x) is -1/6 csc^6(x)+C

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integral of csc^6(x)cot(x)

\int csc^{6} (x) cot (x) d x

- \frac{1}{6} csc^{6} (x) + C

Solution steps

\int csc^{6} (x) cot (x) d x

Rewrite using trig identities

= \int \frac{cos ( x )}{sin ^{7} ( x )} d x

Apply u-substitution

: \int \frac{1}{u ^{7}} d u

= \int \frac{1}{u ^{7}} d u

Apply the Power Rule

: - \frac{1}{6 u ^{6}}

= - \frac{1}{6 u ^{6}}

Substitute back

u = sin (x)

= - \frac{1}{6 sin ^{6} ( x )}

Simplify

- \frac{1}{6 sin ^{6} ( x )} : - \frac{1}{6} csc^{6} (x)

= - \frac{1}{6} csc^{6} (x)

Add a constant to the solution

= - \frac{1}{6} csc^{6} (x) + C

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What is the integral of csc^6(x)cot(x) ?
The integral of csc^6(x)cot(x) is -1/6 csc^6(x)+C