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受欢迎的微积分 >

integral of (x+10)/(x^2+6x+13)

解答

\int \frac{x + 10}{x ^{2} + 6 x + 13} d x

解答

\frac{1}{2} ln x^{2} + 6 x + 13 + \frac{7}{2} arctan (\frac{x + 3}{2}) + C

求解步骤

\int \frac{x + 10}{x ^{2} + 6 x + 13} d x

使用换元积分法

= \int \frac{u}{u ^{2} - 14 u + 53} d u

对

u^{2} - 14 u + 53

配方:

(u - 7)^{2} + 4

= \int \frac{u}{( u - 7 ) ^{2} + 4} d u

使用换元积分法

= \int \frac{v + 7}{v ^{2} + 4} d v

乘开

\frac{v + 7}{v ^{2} + 4} : \frac{v}{v ^{2} + 4} + \frac{7}{v ^{2} + 4}

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} + 4} d v + \int \frac{7}{v ^{2} + 4} d v

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

\int \frac{7}{v ^{2} + 4} d v = \frac{7}{2} arctan (\frac{v}{2})

= \frac{1}{2} ln v^{2} + 4 + \frac{7}{2} arctan (\frac{v}{2})

代回

= \frac{1}{2} ln (x + 10 - 7)^{2} + 4 + \frac{7}{2} arctan (\frac{x + 10 - 7}{2})

化简

\frac{1}{2} ln (x + 10 - 7)^{2} + 4 + \frac{7}{2} arctan (\frac{x + 10 - 7}{2}) : \frac{1}{2} ln x^{2} + 6 x + 13 + \frac{7}{2} arctan (\frac{x + 3}{2})

= \frac{1}{2} ln x^{2} + 6 x + 13 + \frac{7}{2} arctan (\frac{x + 3}{2})

解答补常数

= \frac{1}{2} ln x^{2} + 6 x + 13 + \frac{7}{2} arctan (\frac{x + 3}{2}) + C

作图

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