What is the integral from 0 to 2pi of cos^4(x) ?

The integral from 0 to 2pi of cos^4(x) is (3pi)/4

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integral from 0 to 2pi of cos^4(x)

\int_{0}^{2 π} cos^{4} (x) d x

\frac{3 π}{4}

Decimal Notation

2.35619 \dots

Solution steps

\int_{0}^{2 π} cos^{4} (x) d x

Simplify

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

= \int_{0}^{2 π} 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}^{2 π}

= [cos^{3} (x) sin (x) - \int - 3 cos^{2} (x) sin^{2} (x) d x]_{0}^{2 π}

\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}^{2 π}

Simplify

[cos^{3} (x) sin (x) - (- \frac{3}{8} (x - \frac{1}{4} sin (4 x)))]_{0}^{2 π} : [\frac{1}{32} (32 cos^{3} (x) sin (x) + 3 (4 x - sin (4 x)))]_{0}^{2 π}

= [\frac{1}{32} (32 cos^{3} (x) sin (x) + 3 (4 x - sin (4 x)))]_{0}^{2 π}

Compute the boundaries:

\frac{3 π}{4}

= \frac{3 π}{4}

\int_{0}^{\frac{π}{20}} cos (10 x) d x

n = 1 \sum \infty \frac{2 ^{n}}{n}

\frac{d}{d x} (\frac{x ^{2} + x + 1}{x ^{2} + 1})

\frac{d}{d x} (\frac{1}{2} (tan (x) - x))

n = 0 \sum \infty n^{3} x^{n}

What is the integral from 0 to 2pi of cos^4(x) ?
The integral from 0 to 2pi of cos^4(x) is (3pi)/4