integral from 0 to pi of 12sin^4(x)

解答

\int_{0}^{π} 12 sin^{4} (x) d x

\frac{9 π}{2}

十进制

14.13716 \dots

求解步骤

\int_{0}^{π} 12 sin^{4} (x) d x

提出常数:

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

= 12 \cdot \int_{0}^{π} sin^{4} (x) d x

使用降次积分法

= 12 (- [\frac{cos ( x ) sin ^{3} ( x )}{4}]_{0}^{π} + \frac{3}{4} \cdot \int_{0}^{π} sin^{2} (x) d x)

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

= 12 (- [\frac{cos ( x ) sin ^{3} ( x )}{4}]_{0}^{π} + \frac{3}{4} \cdot \frac{π}{2})

化简

12 (- [\frac{cos ( x ) sin ^{3} ( x )}{4}]_{0}^{π} + \frac{3}{4} \cdot \frac{π}{2}) : 12 (- [\frac{1}{4} sin^{3} (x) cos (x)]_{0}^{π} + \frac{3 π}{8})

= 12 (- [\frac{1}{4} sin^{3} (x) cos (x)]_{0}^{π} + \frac{3 π}{8})

带入上下限计算后得:

0

= 12 (- 0 + \frac{3 π}{8})

化简

= \frac{9 π}{2}

\int_{1}^{2} (7 x + x^{2}) d x

d er i v a t i v e cos (2^{x})

x^{2} y \frac{d y}{d x} = e^{y}

x \to 2 - lim (\frac{x - 2}{∣ x - 2 ∣})

y^{^{'}} = \frac{( 3 x ^{2} - e ^{x} )}{2 y - 5}