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受欢迎的微积分 >

integral of (2x-3)/(x^2+6x+13)

解答

\int \frac{2 x - 3}{x ^{2} + 6 x + 13} d x

解答

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

求解步骤

\int \frac{2 x - 3}{x ^{2} + 6 x + 13} d x

对

x^{2} + 6 x + 13

配方:

(x + 3)^{2} + 4

= \int \frac{2 x - 3}{( x + 3 ) ^{2} + 4} d x

使用换元积分法

= \int \frac{2 u - 9}{u ^{2} + 4} d u

使用换元积分法

= \int \frac{4 v - 9}{2 ( v ^{2} + 1 )} d v

提出常数:

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

= \frac{1}{2} \cdot \int \frac{4 v - 9}{v ^{2} + 1} d v

乘开

\frac{4 v - 9}{v ^{2} + 1} : \frac{4 v}{v ^{2} + 1} - \frac{9}{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

= \frac{1}{2} (\int \frac{4 v}{v ^{2} + 1} d v - \int \frac{9}{v ^{2} + 1} d v)

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

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

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

代回

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

化简

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

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

解答补常数

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

作图

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