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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

对

x^{2} + 2 x + 3

配方:

(x + 1)^{2} + 2

= \int \frac{x + 4}{( x + 1 ) ^{2} + 2} d x

使用换元积分法

= \int \frac{u + 3}{u ^{2} + 2} d u

使用换元积分法

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

乘开

\frac{2 v + 3}{v ^{2} + 1} : \frac{2 v}{v ^{2} + 1} + \frac{3}{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{2 v}{v ^{2} + 1} d v + \int \frac{3}{v ^{2} + 1} d v)

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

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

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

代回

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

化简

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

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

解答补常数

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

作图

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