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受欢迎的微积分 >

integral of (8x^3+3)/((x^2+2)^2)

解答

\int \frac{8 x ^{3} + 3}{( x ^{2} + 2 ) ^{2}} d x

解答

2 (22 ln \frac{x ^{2}}{2} + 1 + \frac{1}{4} (\frac{16 2}{x ^{2} + 2} + \frac{3}{2} (arctan (\frac{x}{2}) + \frac{2 x}{2 + x ^{2}}))) + C

求解步骤

\int \frac{8 x ^{3} + 3}{( x ^{2} + 2 ) ^{2}} d x

使用换元积分法

= \int \frac{2 ( 16 2 u ^{3} + 3 )}{( 2 u ^{2} + 2 ) ^{2}} d u

提出常数:

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

= 2 \cdot \int \frac{16 2 u ^{3} + 3}{( 2 u ^{2} + 2 ) ^{2}} d u

将

\frac{16 2 u ^{3} + 3}{( 2 u ^{2} + 2 ) ^{2}}

用部份分式展开:

\frac{4 2 u}{u ^{2} + 1} + \frac{- 16 2 u + 3}{4 ( u ^{2} + 1 ) ^{2}}

= 2 \cdot \int \frac{4 2 u}{u ^{2} + 1} + \frac{- 16 2 u + 3}{4 ( u ^{2} + 1 ) ^{2}} d u

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{4 2 u}{u ^{2} + 1} d u + \int \frac{- 16 2 u + 3}{4 ( u ^{2} + 1 ) ^{2}} d u)

\int \frac{4 2 u}{u ^{2} + 1} d u = 22 ln u^{2} + 1

\int \frac{- 16 2 u + 3}{4 ( u ^{2} + 1 ) ^{2}} d u = \frac{1}{4} (\frac{8 2}{u ^{2} + 1} + \frac{3}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

= 2 (22 ln u^{2} + 1 + \frac{1}{4} (\frac{8 2}{u ^{2} + 1} + \frac{3}{2} (arctan (u) + \frac{u}{1 + u ^{2}})))

u = \frac{x}{2}

代回

= 2 22 ln (\frac{x}{2})^{2} + 1 + \frac{1}{4} \frac{8 2}{( \frac{x}{2} ) ^{2} + 1} + \frac{3}{2} arctan (\frac{x}{2}) + \frac{\frac{x}{2}}{1 + ( \frac{x}{2} ) ^{2}}

化简

2 22 ln (\frac{x}{2})^{2} + 1 + \frac{1}{4} \frac{8 2}{( \frac{x}{2} ) ^{2} + 1} + \frac{3}{2} arctan (\frac{x}{2}) + \frac{\frac{x}{2}}{1 + ( \frac{x}{2} ) ^{2}} : 2 (22 ln \frac{x ^{2}}{2} + 1 + \frac{1}{4} (\frac{16 2}{x ^{2} + 2} + \frac{3}{2} (arctan (\frac{x}{2}) + \frac{2 x}{2 + x ^{2}})))

= 2 (22 ln \frac{x ^{2}}{2} + 1 + \frac{1}{4} (\frac{16 2}{x ^{2} + 2} + \frac{3}{2} (arctan (\frac{x}{2}) + \frac{2 x}{2 + x ^{2}})))

解答补常数

= 2 (22 ln \frac{x ^{2}}{2} + 1 + \frac{1}{4} (\frac{16 2}{x ^{2} + 2} + \frac{3}{2} (arctan (\frac{x}{2}) + \frac{2 x}{2 + x ^{2}}))) + C

作图

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