integral of (-10x^2)/((x+1)^3)

解答

\int \frac{- 10 x ^{2}}{( x + 1 ) ^{3}} d x

- 10 (\frac{x ^{3}}{( x + 1 ) ^{3}} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}) + C

求解步骤

\int \frac{- 10 x ^{2}}{( x + 1 ) ^{3}} d x

提出常数:

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

= - 10 \cdot \int \frac{x ^{2}}{( x + 1 ) ^{3}} d x

使用分布积分法

= - 10 (\frac{x ^{3}}{( x + 1 ) ^{3}} - \int \frac{x ^{2} ( - x + 2 )}{( x + 1 ) ^{4}} d x)

\int \frac{x ^{2} ( - x + 2 )}{( x + 1 ) ^{4}} d x = - ln ∣ x + 1 ∣ - \frac{3}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - \frac{1}{3 ( x + 1 ) ^{3}} - \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} + \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}

= - 10 (\frac{x ^{3}}{( x + 1 ) ^{3}} - (- ln ∣ x + 1 ∣ - \frac{3}{x + 1} + \frac{3}{2 ( x + 1 ) ^{2}} - \frac{1}{3 ( x + 1 ) ^{3}} - \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} + \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}))

化简

= - 10 (\frac{x ^{3}}{( x + 1 ) ^{3}} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}})

解答补常数

= - 10 (\frac{x ^{3}}{( x + 1 ) ^{3}} + ln ∣ x + 1 ∣ + \frac{3}{x + 1} - \frac{3}{2 ( x + 1 ) ^{2}} + \frac{1}{3 ( x + 1 ) ^{3}} + \frac{2 x ^{2}}{3 ( x + 1 ) ^{3}} - \frac{2 ( - 2 x - 1 )}{3 ( x + 1 ) ^{2}}) + C

\int x (x + 4)^{2} d x

\int \frac{x ^{2}}{7 + x} d x

\int e^{s x} d x

\int x^{2} sec^{2} (x^{3}) d x

\int_{0}^{\infty} t \cdot e^{- s t} d t