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受欢迎的微积分 >

integral of (13)/((x-2)(x^2+9))

解答

\int \frac{13}{( x - 2 ) ( x ^{2} + 9 )} d x

解答

ln ∣ x - 2 ∣ + \frac{1}{6} (- 3 ln \frac{x ^{2}}{9} + 1 - 4 arctan (\frac{x}{3})) + C

求解步骤

\int \frac{13}{( x - 2 ) ( x ^{2} + 9 )} d x

提出常数:

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

= 13 \cdot \int \frac{1}{( x - 2 ) ( x ^{2} + 9 )} d x

使用换元积分法

= 13 \cdot \int \frac{1}{3 ( 3 u - 2 ) ( u ^{2} + 1 )} d u

提出常数:

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

= 13 \cdot \frac{1}{3} \cdot \int \frac{1}{( 3 u - 2 ) ( u ^{2} + 1 )} d u

将

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

用部份分式展开:

\frac{9}{13 ( 3 u - 2 )} + \frac{- 3 u - 2}{13 ( u ^{2} + 1 )}

= 13 \cdot \frac{1}{3} \cdot \int \frac{9}{13 ( 3 u - 2 )} + \frac{- 3 u - 2}{13 ( u ^{2} + 1 )} 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

= 13 \cdot \frac{1}{3} (\int \frac{9}{13 ( 3 u - 2 )} d u + \int \frac{- 3 u - 2}{13 ( u ^{2} + 1 )} d u)

\int \frac{9}{13 ( 3 u - 2 )} d u = \frac{3}{13} ln ∣ 3 u - 2 ∣

\int \frac{- 3 u - 2}{13 ( u ^{2} + 1 )} d u = \frac{1}{13} (- \frac{3}{2} ln u^{2} + 1 - 2 arctan (u))

= 13 \cdot \frac{1}{3} (\frac{3}{13} ln ∣ 3 u - 2 ∣ + \frac{1}{13} (- \frac{3}{2} ln u^{2} + 1 - 2 arctan (u)))

u = \frac{x}{3}

代回

= 13 \cdot \frac{1}{3} (\frac{3}{13} ln 3 \cdot \frac{x}{3} - 2 + \frac{1}{13} (- \frac{3}{2} ln (\frac{x}{3})^{2} + 1 - 2 arctan (\frac{x}{3})))

化简

13 \cdot \frac{1}{3} (\frac{3}{13} ln 3 \cdot \frac{x}{3} - 2 + \frac{1}{13} (- \frac{3}{2} ln (\frac{x}{3})^{2} + 1 - 2 arctan (\frac{x}{3}))) : ln ∣ x - 2 ∣ + \frac{1}{6} (- 3 ln \frac{x ^{2}}{9} + 1 - 4 arctan (\frac{x}{3}))

= ln ∣ x - 2 ∣ + \frac{1}{6} (- 3 ln \frac{x ^{2}}{9} + 1 - 4 arctan (\frac{x}{3}))

解答补常数

= ln ∣ x - 2 ∣ + \frac{1}{6} (- 3 ln \frac{x ^{2}}{9} + 1 - 4 arctan (\frac{x}{3})) + C

作图

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