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受欢迎的微积分 >

integral of x/(2x^2+12x+23)

解答

\int \frac{x}{2 x ^{2} + 12 x + 23} d x

解答

\frac{1}{4} ln 2 x^{2} + 12 x + 23 - \frac{3}{10} arctan (\frac{2}{5} (x + 3)) + C

求解步骤

\int \frac{x}{2 x ^{2} + 12 x + 23} d x

对

2 x^{2} + 12 x + 23

配方:

2 (x + 3)^{2} + 5

= \int \frac{x}{2 ( x + 3 ) ^{2} + 5} d x

使用换元积分法

= \int \frac{u - 3}{2 u ^{2} + 5} d u

乘开

\frac{u - 3}{2 u ^{2} + 5} : \frac{u}{2 u ^{2} + 5} - \frac{3}{2 u ^{2} + 5}

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} + 5} d u - \int \frac{3}{2 u ^{2} + 5} d u

\int \frac{u}{2 u ^{2} + 5} d u = \frac{1}{4} ln 2 u^{2} + 5

\int \frac{3}{2 u ^{2} + 5} d u = \frac{3}{10} arctan (\frac{2}{5} u)

= \frac{1}{4} ln 2 u^{2} + 5 - \frac{3}{10} arctan (\frac{2}{5} u)

u = x + 3

代回

= \frac{1}{4} ln 2 (x + 3)^{2} + 5 - \frac{3}{10} arctan (\frac{2}{5} (x + 3))

化简

\frac{1}{4} ln 2 (x + 3)^{2} + 5 - \frac{3}{10} arctan (\frac{2}{5} (x + 3)) : \frac{1}{4} ln 2 x^{2} + 12 x + 23 - \frac{3}{10} arctan (\frac{2}{5} (x + 3))

= \frac{1}{4} ln 2 x^{2} + 12 x + 23 - \frac{3}{10} arctan (\frac{2}{5} (x + 3))

解答补常数

= \frac{1}{4} ln 2 x^{2} + 12 x + 23 - \frac{3}{10} arctan (\frac{2}{5} (x + 3)) + C

作图

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