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

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

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integral of (cos^2(x))^2

integral

(cos^{2} (x))^{2}

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

Solution steps

\int (cos^{2} (x))^{2} d x

Simplify

(cos^{2} (x))^{2} : cos^{4} (x)

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

Simplify

cos^{4} (x) : cos^{3} (x) cos (x)

= \int cos^{3} (x) cos (x) d x

Apply Integration By Parts

: cos^{3} (x) sin (x) - \int - 3 cos^{2} (x) sin^{2} (x) d x

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

\int - 3 cos^{2} (x) sin^{2} (x) d x = - \frac{3}{8} (x - \frac{1}{4} sin (4 x))

= cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))

Simplify

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

Add a constant to the solution

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

\frac{1}{( x ^{n} )}

x^{2} (e^{x})^{3}

n^{2}

1 + 16 x^{2}

\frac{( e ^{2 x} )}{x}

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