What is the integral of (sin^4(x))/(cos^2(x)) ?

The integral of (sin^4(x))/(cos^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 (sin^4(x))/(cos^2(x))

integral

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

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