What is the integral from 0 to 2pi of (cos(x))^6 ?

The integral from 0 to 2pi of (cos(x))^6 is (5pi)/8

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 2pi of (cos(x))^6

\int_{0}^{2 π} (cos (x))^{6} d x

\frac{5 π}{8}

Decimal

1.96349 \dots

Solution steps

\int_{0}^{2 π} (cos (x))^{6} d x

Simplify

cos^{6} (x) : cos^{5} (x) cos (x)

= \int_{0}^{2 π} cos^{5} (x) cos (x) d x

Simplify

cos^{5} (x) cos (x) : cos^{6} (x)

= \int_{0}^{2 π} cos^{6} (x) d x

Apply Integral Reduction

= [\frac{sin ( x ) cos ^{5} ( x )}{6}]_{0}^{2 π} + \frac{5}{6} \cdot \int_{0}^{2 π} cos^{4} (x) d x

\int_{0}^{2 π} cos^{4} (x) d x = \frac{3 π}{4}

= [\frac{sin ( x ) cos ^{5} ( x )}{6}]_{0}^{2 π} + \frac{5}{6} \cdot \frac{3 π}{4}

Simplify

[\frac{sin ( x ) cos ^{5} ( x )}{6}]_{0}^{2 π} + \frac{5}{6} \cdot \frac{3 π}{4} : [\frac{1}{6} cos^{5} (x) sin (x)]_{0}^{2 π} + \frac{5 π}{8}

= [\frac{1}{6} cos^{5} (x) sin (x)]_{0}^{2 π} + \frac{5 π}{8}

Compute the boundaries:

0

= 0 + \frac{5 π}{8}

Simplify

= \frac{5 π}{8}

\int_{0}^{π} 11 x sin (x) d x

\int_{8}^{11} x d x

\int_{0}^{3} \frac{t}{7 + t ^{2}} d t

\int_{- 2}^{1} x (x - 1) (x + 2) d x

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

What is the integral from 0 to 2pi of (cos(x))^6 ?
The integral from 0 to 2pi of (cos(x))^6 is (5pi)/8