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受欢迎的微积分 >

integral of (x+7)/(x^2+4x+8)

解答

\int \frac{x + 7}{x ^{2} + 4 x + 8} d x

解答

\frac{1}{2} ln x^{2} + 4 x + 8 + \frac{5}{2} arctan (\frac{x + 2}{2}) + C

求解步骤

\int \frac{x + 7}{x ^{2} + 4 x + 8} d x

使用换元积分法

= \int \frac{u}{u ^{2} - 10 u + 29} d u

对

u^{2} - 10 u + 29

配方:

(u - 5)^{2} + 4

= \int \frac{u}{( u - 5 ) ^{2} + 4} d u

使用换元积分法

= \int \frac{v + 5}{v ^{2} + 4} d v

乘开

\frac{v + 5}{v ^{2} + 4} : \frac{v}{v ^{2} + 4} + \frac{5}{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{5}{v ^{2} + 4} d v

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

\int \frac{5}{v ^{2} + 4} d v = \frac{5}{2} arctan (\frac{v}{2})

= \frac{1}{2} ln v^{2} + 4 + \frac{5}{2} arctan (\frac{v}{2})

代回

= \frac{1}{2} ln (x + 7 - 5)^{2} + 4 + \frac{5}{2} arctan (\frac{x + 7 - 5}{2})

化简

\frac{1}{2} ln (x + 7 - 5)^{2} + 4 + \frac{5}{2} arctan (\frac{x + 7 - 5}{2}) : \frac{1}{2} ln x^{2} + 4 x + 8 + \frac{5}{2} arctan (\frac{x + 2}{2})

= \frac{1}{2} ln x^{2} + 4 x + 8 + \frac{5}{2} arctan (\frac{x + 2}{2})

解答补常数

= \frac{1}{2} ln x^{2} + 4 x + 8 + \frac{5}{2} arctan (\frac{x + 2}{2}) + C

作图

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x \to \infty lim (\frac{( e ^{2 x} )}{x ^{2}})

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