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

解答

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

cosh (x) + C

求解步骤

\int \frac{e ^{x} - e ^{- x}}{2} d x

提出常数:

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

= \frac{1}{2} \cdot \int e^{x} - e^{- x} d x

使用三角恒等式改写

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

提出常数:

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

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

使用积分常见定则:

\int sinh (x) d x = cosh (x)

= \frac{1}{2} \cdot 2 cosh (x)

化简

\frac{1}{2} \cdot 2 cosh (x) : cosh (x)

= cosh (x)

解答补常数

= cosh (x) + C

\int \frac{\frac{1}{e ^{x} + e ^{2 x}}}{- e ^{- 3 x}} d x

x \to 2 - lim (\frac{x - 1}{∣ x - 1 ∣})

\frac{d}{d x} (3 x^{3} y^{2})

\frac{d y}{d x} = \frac{2 x + sec ^{2} ( x )}{4 y}

\frac{\partial}{\partial y} (y ln (x))