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

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integral of (cos^4(x))

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

= 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_{0}^{4.5} 130 x d x

\int_{π}^{2 π} sin^{2} (x) d x

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

\frac{\partial}{\partial y} (2 x^{\frac{1}{2}} y^{\frac{1}{4}} + 7)

\int \frac{7 x ^{2} + x + 7}{( x ^{2} + 1 ) ^{2}} 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