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受欢迎的微积分 >

integral of x/(2x^2+8x+11)

解答

\int \frac{x}{2 x ^{2} + 8 x + 11} d x

解答

\frac{1}{4} ln 2 x^{2} + 8 x + 11 - \frac{2}{3} arctan (\frac{2}{3} (x + 2)) + C

求解步骤

\int \frac{x}{2 x ^{2} + 8 x + 11} d x

对

2 x^{2} + 8 x + 11

配方:

2 (x + 2)^{2} + 3

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

使用换元积分法

= \int \frac{u - 2}{2 u ^{2} + 3} d u

乘开

\frac{u - 2}{2 u ^{2} + 3} : \frac{u}{2 u ^{2} + 3} - \frac{2}{2 u ^{2} + 3}

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 \frac{u}{2 u ^{2} + 3} d u - \int \frac{2}{2 u ^{2} + 3} d u

\int \frac{u}{2 u ^{2} + 3} d u = \frac{1}{4} ln 2 u^{2} + 3

\int \frac{2}{2 u ^{2} + 3} d u = \frac{2}{3} arctan (\frac{2}{3} u)

= \frac{1}{4} ln 2 u^{2} + 3 - \frac{2}{3} arctan (\frac{2}{3} u)

u = x + 2

代回

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

化简

\frac{1}{4} ln 2 (x + 2)^{2} + 3 - \frac{2}{3} arctan (\frac{2}{3} (x + 2)) : \frac{1}{4} ln 2 x^{2} + 8 x + 11 - \frac{2}{3} arctan (\frac{2}{3} (x + 2))

= \frac{1}{4} ln 2 x^{2} + 8 x + 11 - \frac{2}{3} arctan (\frac{2}{3} (x + 2))

解答补常数

= \frac{1}{4} ln 2 x^{2} + 8 x + 11 - \frac{2}{3} arctan (\frac{2}{3} (x + 2)) + C

作图

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