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受欢迎的微积分 >

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

解答

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

解答

ln x^{2} + 2 x + 2 - 2 arctan (x + 1) + C

求解步骤

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

对

x^{2} + 2 x + 2

配方:

(x + 1)^{2} + 1

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

使用换元积分法

= 2 \cdot \int \frac{u - 1}{u ^{2} + 1} d u

乘开

\frac{u - 1}{u ^{2} + 1} : \frac{u}{u ^{2} + 1} - \frac{1}{u ^{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 (\int \frac{u}{u ^{2} + 1} d u - \int \frac{1}{u ^{2} + 1} d u)

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

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

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

u = x + 1

代回

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

化简

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

= ln x^{2} + 2 x + 2 - 2 arctan (x + 1)

解答补常数

= ln x^{2} + 2 x + 2 - 2 arctan (x + 1) + C

作图

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