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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

对

x^{2} - 2 x + 3

配方:

(x - 1)^{2} + 2

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

使用换元积分法

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

乘开

\frac{u + 1}{u ^{2} + 2} : \frac{u}{u ^{2} + 2} + \frac{1}{u ^{2} + 2}

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

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

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

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

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

u = x - 1

代回

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

化简

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

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

解答补常数

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

作图

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