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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 sinh (x))^{2} d x

化简

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

= \int 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 sinh^{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

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limit as x approaches 3 of x+5

x \to 3 lim (x + 5)

(\partial)/(\partial y)(sin(8x)cos(3y))

\frac{\partial}{\partial y} (sin (8 x) cos (3 y))

derivative (6x^2-9x)e^x

d er i v a t i v e (6 x^{2} - 9 x) e^{x}

integral of (t^3)/m-(cos(t))/m

\int \frac{t ^{3}}{m} - \frac{cos ( t )}{m} d t

derivative of (x^2-9x+20/((x+2)(x-2)))

\frac{d}{d x} (\frac{x ^{2} - 9 x + 20}{( x + 2 ) ( x - 2 )})