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受欢迎的微积分 >

integral of (6x+11)/(x^2+2x+5)

解答

\int \frac{6 x + 11}{x ^{2} + 2 x + 5} d x

解答

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

求解步骤

\int \frac{6 x + 11}{x ^{2} + 2 x + 5} d x

对

x^{2} + 2 x + 5

配方:

(x + 1)^{2} + 4

= \int \frac{6 x + 11}{( x + 1 ) ^{2} + 4} d x

使用换元积分法

= \int \frac{6 u + 5}{u ^{2} + 4} d u

乘开

\frac{6 u + 5}{u ^{2} + 4} : \frac{6 u}{u ^{2} + 4} + \frac{5}{u ^{2} + 4}

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{6 u}{u ^{2} + 4} d u + \int \frac{5}{u ^{2} + 4} d u

\int \frac{6 u}{u ^{2} + 4} d u = 3 ln u^{2} + 4

\int \frac{5}{u ^{2} + 4} d u = \frac{5}{2} arctan (\frac{u}{2})

= 3 ln u^{2} + 4 + \frac{5}{2} arctan (\frac{u}{2})

u = x + 1

代回

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

化简

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

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

解答补常数

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

作图

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