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受欢迎的微积分 >

integral from-6 to 6 of (e^x-e^{-x})^2

解答

\int_{- 6}^{6} (e^{x} - e^{- x})^{2} d x

解答

\frac{e ^{24} - 1}{e ^{12}} - 24

+1

十进制

162730.79141 \dots

求解步骤

\int_{- 6}^{6} (e^{x} - e^{- x})^{2} d x

使用三角恒等式改写

= \int_{- 6}^{6} (2 sinh (x))^{2} d x

化简

(2 sinh (x))^{2} : 4 sinh^{2} (x)

= \int_{- 6}^{6} 4 sinh^{2} (x) d x

提出常数:

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

= 4 \cdot \int_{- 6}^{6} sinh^{2} (x) d x

使用三角恒等式改写

= 4 \cdot \int_{- 6}^{6} \frac{- 1 + cosh ( 2 x )}{2} d x

提出常数:

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

= 4 \cdot \frac{1}{2} \cdot \int_{- 6}^{6} - 1 + cosh (2 x) d x

使用积分加法定则:

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

= 4 \cdot \frac{1}{2} (- \int_{- 6}^{6} 1 d x + \int_{- 6}^{6} cosh (2 x) d x)

\int_{- 6}^{6} 1 d x = 12

\int_{- 6}^{6} cosh (2 x) d x = \frac{e ^{24} - 1}{2 e ^{12}}

= 4 \cdot \frac{1}{2} (- 12 + \frac{e ^{24} - 1}{2 e ^{12}})

化简

4 \cdot \frac{1}{2} (- 12 + \frac{e ^{24} - 1}{2 e ^{12}}) : \frac{e ^{24} - 1}{e ^{12}} - 24

= \frac{e ^{24} - 1}{e ^{12}} - 24

作图

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d er i v a t i v e \frac{cos ( t )}{t ^{7}}

(\partial)/(\partial b)((ab)/(2c+15))

\frac{\partial}{\partial b} (\frac{ab}{2 c + 15})

limit as x approaches 0 of xe^{2x}

x \to 0 lim (x e^{2 x})

面积 y=3(x^3-x),y=0

a re a y = 3 (x^{3} - x), y = 0

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\int_{- 1}^{2} (2 x - 1) (4 x + 5) d x