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

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

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

integral

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

\frac{1}{2} (sin^{3} (x) cos (x) - \frac{3}{2} x + \frac{3}{4} sin (2 x) + sin^{4} (x) tan (x)) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int sin^{2} (x) tan^{2} (x) d x

Rewrite using trig identities

= \frac{1}{2} \cdot \int \frac{sin ^{4} ( x )}{cos ^{2} ( x )} d x

Apply Integration By Parts

= \frac{1}{2} (- \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)

= \frac{1}{2} (- (- sin^{3} (x) cos (x) + \frac{3}{2} x - \frac{3}{4} sin (2 x)) + sin^{4} (x) tan (x))

Simplify

= \frac{1}{2} (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

= \frac{1}{2} (sin^{3} (x) cos (x) - \frac{3}{2} x + \frac{3}{4} sin (2 x) + sin^{4} (x) tan (x)) + C

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