What is the integral of (sin^2(y))/(tan^4(y)) ?

The integral of (sin^2(y))/(tan^4(y)) is -cos^4(y)cot(y)-4(cos^3(y)sin(y)+3/8 (y-1/4 sin(4y)))+C

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integral of (sin^2(y))/(tan^4(y))

\int \frac{sin ^{2} ( y )}{tan ^{4} ( y )} d y

- cos^{4} (y) cot (y) - 4 (cos^{3} (y) sin (y) + \frac{3}{8} (y - \frac{1}{4} sin (4 y))) + C

Solution steps

\int \frac{sin ^{2} ( y )}{tan ^{4} ( y )} d y

Rewrite using trig identities

= \int \frac{cos ^{4} ( y )}{sin ^{2} ( y )} d y

Apply Integration By Parts

: - cos^{4} (y) cot (y) - \int 4 cos^{3} (y) sin (y) cot (y) d y

= - cos^{4} (y) cot (y) - \int 4 cos^{3} (y) sin (y) cot (y) d y

\int 4 cos^{3} (y) sin (y) cot (y) d y = 4 (cos^{3} (y) sin (y) + \frac{3}{8} (y - \frac{1}{4} sin (4 y)))

= - cos^{4} (y) cot (y) - 4 (cos^{3} (y) sin (y) + \frac{3}{8} (y - \frac{1}{4} sin (4 y)))

Add a constant to the solution

= - cos^{4} (y) cot (y) - 4 (cos^{3} (y) sin (y) + \frac{3}{8} (y - \frac{1}{4} sin (4 y))) + C

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What is the integral of (sin^2(y))/(tan^4(y)) ?
The integral of (sin^2(y))/(tan^4(y)) is -cos^4(y)cot(y)-4(cos^3(y)sin(y)+3/8 (y-1/4 sin(4y)))+C