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受欢迎的微积分 >

integral of (x+3)/(x^2+4x+13)

解答

\int \frac{x + 3}{x ^{2} + 4 x + 13} d x

解答

\frac{1}{2} ln x^{2} + 4 x + 13 + \frac{1}{3} arctan (\frac{x + 2}{3}) + C

求解步骤

\int \frac{x + 3}{x ^{2} + 4 x + 13} d x

使用换元积分法

= \int \frac{u}{u ^{2} - 2 u + 10} d u

对

u^{2} - 2 u + 10

配方:

(u - 1)^{2} + 9

= \int \frac{u}{( u - 1 ) ^{2} + 9} d u

使用换元积分法

= \int \frac{v + 1}{v ^{2} + 9} d v

乘开

\frac{v + 1}{v ^{2} + 9} : \frac{v}{v ^{2} + 9} + \frac{1}{v ^{2} + 9}

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{v}{v ^{2} + 9} d v + \int \frac{1}{v ^{2} + 9} d v

\int \frac{v}{v ^{2} + 9} d v = \frac{1}{2} ln v^{2} + 9

\int \frac{1}{v ^{2} + 9} d v = \frac{1}{3} arctan (\frac{v}{3})

= \frac{1}{2} ln v^{2} + 9 + \frac{1}{3} arctan (\frac{v}{3})

代回

= \frac{1}{2} ln (x + 3 - 1)^{2} + 9 + \frac{1}{3} arctan (\frac{x + 3 - 1}{3})

化简

\frac{1}{2} ln (x + 3 - 1)^{2} + 9 + \frac{1}{3} arctan (\frac{x + 3 - 1}{3}) : \frac{1}{2} ln x^{2} + 4 x + 13 + \frac{1}{3} arctan (\frac{x + 2}{3})

= \frac{1}{2} ln x^{2} + 4 x + 13 + \frac{1}{3} arctan (\frac{x + 2}{3})

解答补常数

= \frac{1}{2} ln x^{2} + 4 x + 13 + \frac{1}{3} arctan (\frac{x + 2}{3}) + C

作图

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