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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

使用换元积分法

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

提出常数:

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

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

乘开

\frac{4 u - 3}{u ^{2} + 1} : \frac{4 u}{u ^{2} + 1} - \frac{3}{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

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

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

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

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

u = \frac{x}{2}

代回

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

化简

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

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

解答补常数

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

作图

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