What is the integral of (cot(x)csc(x))/(csc^7(x)) ?

The integral of (cot(x)csc(x))/(csc^7(x)) is (sin^6(x))/6+C

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integral of (cot(x)csc(x))/(csc^7(x))

\int \frac{cot ( x ) csc ( x )}{csc ^{7} ( x )} d x

\frac{sin ^{6} ( x )}{6} + C

Solution steps

\int \frac{cot ( x ) csc ( x )}{csc ^{7} ( x )} d x

Rewrite using trig identities

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

Apply u-substitution

: \int u^{5} d u

= \int u^{5} d u

Apply the Power Rule

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

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

Substitute back

u = sin (x)

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

Add a constant to the solution

= \frac{sin ^{6} ( x )}{6} + C

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What is the integral of (cot(x)csc(x))/(csc^7(x)) ?
The integral of (cot(x)csc(x))/(csc^7(x)) is (sin^6(x))/6+C