What is the integral of tan^2(x)sin^2(x) ?

The integral of tan^2(x)sin^2(x) is sin^3(x)cos(x)-3/2 x+3/4 sin(2x)+sin^4(x)tan(x)+C

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integral of tan^2(x)sin^2(x)

\int tan^{2} (x) sin^{2} (x) d x

sin^{3} (x) cos (x) - \frac{3}{2} x + \frac{3}{4} sin (2 x) + sin^{4} (x) tan (x) + C

Solution steps

\int tan^{2} (x) sin^{2} (x) d x

Rewrite using trig identities

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

Apply Integration By Parts

: - \int 4 sin^{4} (x) d x + sin^{4} (x) tan (x)

= - \int 4 sin^{4} (x) d x + sin^{4} (x) tan (x)

\int 4 sin^{4} (x) d x = - sin^{3} (x) cos (x) + \frac{3}{2} x - \frac{3}{4} sin (2 x)

= - (- sin^{3} (x) cos (x) + \frac{3}{2} x - \frac{3}{4} sin (2 x)) + sin^{4} (x) tan (x)

Simplify

= sin^{3} (x) cos (x) - \frac{3}{2} x + \frac{3}{4} sin (2 x) + sin^{4} (x) tan (x)

Add a constant to the solution

= sin^{3} (x) cos (x) - \frac{3}{2} x + \frac{3}{4} sin (2 x) + sin^{4} (x) tan (x) + C

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