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受欢迎的微积分 >

integral of (e^x+e^{-x})^2

解答

\int (e^{x} + e^{- x})^{2} d x

解答

2 x + sinh (2 x) + C

求解步骤

\int (e^{x} + e^{- x})^{2} d x

使用三角恒等式改写

= \int (2 cosh (x))^{2} d x

化简

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

= \int 4 cosh^{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 cosh^{2} (x) d x

使用三角恒等式改写

= 4 \cdot \int \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 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 1 d x + \int cosh (2 x) d x)

\int 1 d x = x

\int cosh (2 x) d x = \frac{1}{2} sinh (2 x)

= 4 \cdot \frac{1}{2} (x + \frac{1}{2} sinh (2 x))

化简

4 \cdot \frac{1}{2} (x + \frac{1}{2} sinh (2 x)) : 2 x + sinh (2 x)

= 2 x + sinh (2 x)

解答补常数

= 2 x + sinh (2 x) + C

作图

流行的例子

integral from 0 to pi/3 of 6sec^3(x)

\int_{0}^{\frac{π}{3}} 6 sec^{3} (x) d x

limit as x approaches-0.001 of (e^x-1)/x

x \to - 0.001 lim (\frac{e ^{x} - 1}{x})

integral from-2 to 3 of x^3

\int_{- 2}^{3} x^{3} d x

integral of 1/(sqrt((x^2-1)^3))

\int \frac{1}{( x ^{2} - 1 ) ^{3}} d x

limit as x approaches 3 of 4x^2

x \to 3 lim (4 x^{2})