What is the integral from 0 to 1 of cos^4(x) ?

The integral from 0 to 1 of cos^4(x) is 3/8 (-1/4 sin(4)+1)+cos^3(1)sin(1)

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integral from 0 to 1 of cos^4(x)

\int_{0}^{1} cos^{4} (x) d x

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

Decimal

0.57867 \dots

Solution steps

\int_{0}^{1} cos^{4} (x) d x

Simplify

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

= \int_{0}^{1} 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]_{0}^{1}

\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)))]_{0}^{1}

Simplify

= [cos^{3} (x) sin (x) + \frac{3}{8} (x - \frac{1}{4} sin (4 x))]_{0}^{1}

Compute the boundaries:

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

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

What is the integral from 0 to 1 of cos^4(x) ?
The integral from 0 to 1 of cos^4(x) is 3/8 (-1/4 sin(4)+1)+cos^3(1)sin(1)