What is the integral of (csc^2(y))/(cot^5(y)) ?

The integral of (csc^2(y))/(cot^5(y)) is (tan^4(y))/4+C

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integral of (csc^2(y))/(cot^5(y))

\int \frac{csc ^{2} ( y )}{cot ^{5} ( y )} d y

\frac{tan ^{4} ( y )}{4} + C

Solution steps

\int \frac{csc ^{2} ( y )}{cot ^{5} ( y )} d y

Rewrite using trig identities

= \int (1 + tan^{2} (y)) tan^{3} (y) d y

Apply u-substitution

: \int u^{3} d u

= \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= \frac{u ^{4}}{4}

Substitute back

u = tan (y)

= \frac{tan ^{4} ( y )}{4}

Add a constant to the solution

= \frac{tan ^{4} ( y )}{4} + C

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What is the integral of (csc^2(y))/(cot^5(y)) ?
The integral of (csc^2(y))/(cot^5(y)) is (tan^4(y))/4+C