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受欢迎的微积分 >

integral from 0 to pi/7 of sin^4(7x)

解答

\int_{0}^{\frac{π}{7}} sin^{4} (7 x) d x

解答

\frac{3 π}{56}

+1

十进制

0.16829 \dots

求解步骤

\int_{0}^{\frac{π}{7}} sin^{4} (7 x) d x

使用换元积分法

= \int_{0}^{π} sin^{4} (u) \frac{1}{7} d u

提出常数:

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

= \frac{1}{7} \cdot \int_{0}^{π} sin^{4} (u) d u

使用降次积分法

= \frac{1}{7} (- [\frac{cos ( u ) sin ^{3} ( u )}{4}]_{0}^{π} + \frac{3}{4} \cdot \int_{0}^{π} sin^{2} (u) d u)

\int_{0}^{π} sin^{2} (u) d u = \frac{π}{2}

= \frac{1}{7} (- [\frac{cos ( u ) sin ^{3} ( u )}{4}]_{0}^{π} + \frac{3}{4} \cdot \frac{π}{2})

化简

\frac{1}{7} (- [\frac{cos ( u ) sin ^{3} ( u )}{4}]_{0}^{π} + \frac{3}{4} \cdot \frac{π}{2}) : \frac{1}{7} (- [\frac{1}{4} sin^{3} (u) cos (u)]_{0}^{π} + \frac{3 π}{8})

= \frac{1}{7} (- [\frac{1}{4} sin^{3} (u) cos (u)]_{0}^{π} + \frac{3 π}{8})

带入上下限计算后得:

0

= \frac{1}{7} (- 0 + \frac{3 π}{8})

化简

= \frac{3 π}{56}

作图

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integral of sin(x)sqrt(1+cos^2(x))

\int sin (x) 1 + cos^{2} (x) d x

(dP)/(dt)=P(10^{-1}-P*10^{-7}),P(0)=7000

\frac{d P}{d t} = P (1 0^{- 1} - P \cdot 1 0^{- 7}), P (0) = 7000