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

The integral from 0 to pi of (cos(x))^6 is (5pi)/(16)

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

\int_{0}^{π} (cos (x))^{6} d x

\frac{5 π}{16}

Decimal Notation

0.98174 \dots

Solution steps

\int_{0}^{π} (cos (x))^{6} d x

Apply u-substitution

: \int_{1}^{- 1} - \frac{u ^{6}}{1 - u ^{2}} d u

= \int_{1}^{- 1} - \frac{u ^{6}}{1 - u ^{2}} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 1}^{1} - \frac{u ^{6}}{1 - u ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - (- \int_{- 1}^{1} \frac{u ^{6}}{1 - u ^{2}} d u)

Apply Trigonometric Substitution

: \int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{6} (v) d v

= - (- \int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{6} (v) d v)

Apply Integral Reduction

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

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

Apply Integral Reduction

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{2} (v) d v)))

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{2} (v) d v)))

Rewrite using trig identities

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 - cos ( 2 v )}{2} d v)))

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 - cos (2 v) d v)))

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \frac{1}{2} (\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d v - \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 v) d v))))

\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d v = π

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 v) d v = 0

= - (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3}{4} \cdot \frac{1}{2} (π - 0))))

Simplify

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

= - [\frac{1}{6} sin^{5} (v) cos (v)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{1}{4} sin^{3} (v) cos (v)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{3 π}{8})

Compute the boundaries:

0

Compute the boundaries:

0

= - 0 + \frac{5}{6} (- 0 + \frac{3 π}{8})

Simplify

= \frac{5 π}{16}

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What is the integral from 0 to pi of (cos(x))^6 ?
The integral from 0 to pi of (cos(x))^6 is (5pi)/(16)