What is the integral of (cot(x)csc(x))/(csc^3(x)) ?

The integral of (cot(x)csc(x))/(csc^3(x)) is (sin^2(x))/2+C

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integral of (cot(x)csc(x))/(csc^3(x))

\int \frac{cot ( x ) csc ( x )}{csc ^{3} ( x )} d x

\frac{sin ^{2} ( x )}{2} + C

Solution steps

\int \frac{cot ( x ) csc ( x )}{csc ^{3} ( x )} d x

Rewrite using trig identities

= \int cos (x) sin (x) d x

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= \frac{u ^{2}}{2}

Substitute back

u = sin (x)

= \frac{sin ^{2} ( x )}{2}

Add a constant to the solution

= \frac{sin ^{2} ( x )}{2} + C

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What is the integral of (cot(x)csc(x))/(csc^3(x)) ?
The integral of (cot(x)csc(x))/(csc^3(x)) is (sin^2(x))/2+C