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

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

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integral from-pi to pi of cos^4(x)

\int_{- π}^{π} cos^{4} (x) d x

\frac{3 π}{4}

Decimal

2.35619 \dots

Solution steps

\int_{- π}^{π} cos^{4} (x) d x

Simplify

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

= \int_{- π}^{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]_{- π}^{π}

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

Simplify

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

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

Compute the boundaries:

\frac{3 π}{4}

= \frac{3 π}{4}

\int_{0}^{1} x^{3} + x^{2} - 2 x d x

\int_{0}^{3} (x - 3)^{2} d x

\int_{1}^{3} (x + 1)^{4} d x

\int_{2}^{7} \frac{x}{x ^{2} + 1} d x

\int_{- 1}^{1} 2 y d y

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