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受欢迎的微积分 >

integral of x/(sqrt(x^2+4x-5))

解答

\int \frac{x}{x ^{2} + 4 x - 5} d x

解答

x^{2} + 4 x - 5 - 2 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2) + C

求解步骤

\int \frac{x}{x ^{2} + 4 x - 5} d x

对

x^{2} + 4 x - 5

配方:

(x + 2)^{2} - 9

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

使用换元积分法

= \int \frac{u - 2}{u ^{2} - 9} d u

乘开

\frac{u - 2}{u ^{2} - 9} : \frac{u}{u ^{2} - 9} - \frac{2}{u ^{2} - 9}

= \int \frac{u}{u ^{2} - 9} - \frac{2}{u ^{2} - 9} 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

= \int \frac{u}{u ^{2} - 9} d u - \int \frac{2}{u ^{2} - 9} d u

\int \frac{u}{u ^{2} - 9} d u = u^{2} - 9

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

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

u = x + 2

代回

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

化简

(x + 2)^{2} - 9 - 2 ln (\frac{1}{3} (x + 2)^{2} - 9 + x + 2) : x^{2} + 4 x - 5 - 2 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2)

= x^{2} + 4 x - 5 - 2 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2)

解答补常数

= x^{2} + 4 x - 5 - 2 ln (\frac{1}{3} x^{2} + 4 x - 5 + x + 2) + C

作图

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