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integral de 0 a pi/2 de (cos(x))^6

Solução

\int_{0}^{\frac{π}{2}} (cos (x))^{6} d x

\frac{5 π}{32}

Decimal

0.49087 \dots

Passos da solução

\int_{0}^{\frac{π}{2}} (cos (x))^{6} d x

Aplicar integração por substituição

= \int_{1}^{0} - \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_{0}^{1} - \frac{u ^{6}}{1 - u ^{2}} d u

Remover a constante:

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

= - (- \int_{0}^{1} \frac{u ^{6}}{1 - u ^{2}} d u)

Aplicar integração por substituição trigonométrica

= - (- \int_{0}^{\frac{π}{2}} sin^{6} (v) d v)

Aplicar a redução de integrais

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

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

Aplicar a redução de integrais

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

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

Reeecreva usando identidades trigonométricas

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

Remover a constante:

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

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

Aplicar a regra da soma:

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

\int_{0}^{\frac{π}{2}} 1 d v = \frac{π}{2}

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

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

Simplificar

- (- (- [\frac{cos ( v ) sin ^{5} ( v )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{cos ( v ) sin ^{3} ( v )}{4}]_{0}^{\frac{π}{2}} + \frac{3}{4} \cdot \frac{1}{2} (\frac{π}{2} - 0)))) : - [\frac{1}{6} sin^{5} (v) cos (v)]_{0}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{1}{4} sin^{3} (v) cos (v)]_{0}^{\frac{π}{2}} + \frac{3 π}{16})

= - [\frac{1}{6} sin^{5} (v) cos (v)]_{0}^{\frac{π}{2}} + \frac{5}{6} (- [\frac{1}{4} sin^{3} (v) cos (v)]_{0}^{\frac{π}{2}} + \frac{3 π}{16})

Calcular os limites:

0

Calcular os limites:

0

= - 0 + \frac{5}{6} (- 0 + \frac{3 π}{16})

Simplificar

= \frac{5 π}{32}

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