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受欢迎的微积分 >

积分 (2x)/((x+1)^2(x^2+2))

解答

积分

\frac{2 x}{( x + 1 ) ^{2} ( x ^{2} + 2 )}

解答

2 (\frac{1}{9} ln ∣ x + 1 ∣ + \frac{1}{3 ( x + 1 )} + \frac{1}{9} (- \frac{1}{2} ln x^{2} + 2 + 22 arctan (\frac{x}{2}))) + C

求解步骤

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

将

\frac{x}{( x + 1 ) ^{2} ( x ^{2} + 2 )}

用部份分式展开:

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

= 2 \cdot \int \frac{1}{9 ( x + 1 )} - \frac{1}{3 ( x + 1 ) ^{2}} + \frac{- x + 4}{9 ( x ^{2} + 2 )} d x

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{1}{9 ( x + 1 )} d x - \int \frac{1}{3 ( x + 1 ) ^{2}} d x + \int \frac{- x + 4}{9 ( x ^{2} + 2 )} d x)

\int \frac{1}{9 ( x + 1 )} d x = \frac{1}{9} ln ∣ x + 1 ∣

\int \frac{1}{3 ( x + 1 ) ^{2}} d x = - \frac{1}{3 ( x + 1 )}

\int \frac{- x + 4}{9 ( x ^{2} + 2 )} d x = \frac{1}{9} (- \frac{1}{2} ln x^{2} + 2 + 22 arctan (\frac{x}{2}))

= 2 (\frac{1}{9} ln ∣ x + 1 ∣ - (- \frac{1}{3 ( x + 1 )}) + \frac{1}{9} (- \frac{1}{2} ln x^{2} + 2 + 22 arctan (\frac{x}{2})))

化简

= 2 (\frac{1}{9} ln ∣ x + 1 ∣ + \frac{1}{3 ( x + 1 )} + \frac{1}{9} (- \frac{1}{2} ln x^{2} + 2 + 22 arctan (\frac{x}{2})))

解答补常数

= 2 (\frac{1}{9} ln ∣ x + 1 ∣ + \frac{1}{3 ( x + 1 )} + \frac{1}{9} (- \frac{1}{2} ln x^{2} + 2 + 22 arctan (\frac{x}{2}))) + C

作图

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