integral of (e^x-1)^2

解答

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

\frac{1}{2} e^{2 x} - 2 e^{x} + x + C

求解步骤

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

使用换元积分法

= \int \frac{( u - 1 ) ^{2}}{u} d u

乘开

\frac{( u - 1 ) ^{2}}{u} : u - 2 + \frac{1}{u}

= \int u - 2 + \frac{1}{u} d u

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

= \int u d u - \int 2 d u + \int \frac{1}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} - 2 u + ln ∣ u ∣

u = e^{x}

代回

= \frac{( e ^{x} ) ^{2}}{2} - 2 e^{x} + ln ∣ e^{x} ∣

化简

\frac{( e ^{x} ) ^{2}}{2} - 2 e^{x} + ln ∣ e^{x} ∣ : \frac{1}{2} e^{2 x} - 2 e^{x} + x

= \frac{1}{2} e^{2 x} - 2 e^{x} + x

解答补常数

= \frac{1}{2} e^{2 x} - 2 e^{x} + x + C

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

d er i v a t i v e \frac{6}{x ^{5}} - \frac{2}{x}

\frac{\partial}{\partial x} (3 (x + 1)^{2})

\frac{\partial}{\partial x} (z^{3} - x^{2} y)

\int (7 - 3 x - 9 x^{2}) d x