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

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

integral of cos^4(x)

integral

cos^{4} (x)

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

Solution steps

\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

\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

\int \frac{3 x + 1}{x} d x

\int (3 x^{5} - 8 x^{3} + 2) d x

\int \frac{( x ^{3} - x ^{4} ) ^{2}}{x} d x

\int \frac{x}{2} e^{- \frac{3 x}{4}} d x

\int \frac{1}{( 3 + cos ( x ) )} d x

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