integral of (1-x)/(3e^x)

解答

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

\frac{x}{3 e ^{x}} + C

求解步骤

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

提出常数:

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

= \frac{1}{3} \cdot \int \frac{1 - x}{e ^{x}} d x

使用换元积分法

= \frac{1}{3} \cdot \int \frac{1 - ln ( u )}{u ^{2}} d u

使用分布积分法

= \frac{1}{3} (- \frac{1 - ln ( u )}{u} - \int \frac{1}{u ^{2}} d u)

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

= \frac{1}{3} (- \frac{1 - ln ( u )}{u} - (- \frac{1}{u}))

u = e^{x}

代回

= \frac{1}{3} (- \frac{1 - ln ( e ^{x} )}{e ^{x}} - (- \frac{1}{e ^{x}}))

化简

\frac{1}{3} (- \frac{1 - ln ( e ^{x} )}{e ^{x}} - (- \frac{1}{e ^{x}})) : \frac{x}{3 e ^{x}}

= \frac{x}{3 e ^{x}}

解答补常数

= \frac{x}{3 e ^{x}} + C

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

x \to 0 lim (\frac{( 1 + x ) ^{3} - 1}{x})

x \to 0 lim (\frac{1}{sin ( x ^{2} )})

\int \frac{5}{x ( x ^{4} + 1 )} d x

\int \frac{2 x ^{2} + 16}{x ( x - 2 ) ^{3}} d x