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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{x ^{2} - 4 x + 13} d x

对

x^{2} - 4 x + 13

配方:

(x - 2)^{2} + 9

= 2 \cdot \int \frac{x}{( x - 2 ) ^{2} + 9} d x

使用换元积分法

= 2 \cdot \int \frac{u + 2}{u ^{2} + 9} d u

使用换元积分法

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{3} \cdot \int \frac{3 v + 2}{v ^{2} + 1} d v

乘开

\frac{3 v + 2}{v ^{2} + 1} : \frac{3 v}{v ^{2} + 1} + \frac{2}{v ^{2} + 1}

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

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

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

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

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

代回

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

化简

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

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

解答补常数

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

作图

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