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受欢迎的微积分 >

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

解答

\int \frac{( 2 x + 5 )}{x ^{2} + 2 x + 5} d x

解答

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

求解步骤

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

对

x^{2} + 2 x + 5

配方:

(x + 1)^{2} + 4

= \int \frac{2 x + 5}{( x + 1 ) ^{2} + 4} d x

使用换元积分法

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

乘开

\frac{2 u + 3}{u ^{2} + 4} : \frac{2 u}{u ^{2} + 4} + \frac{3}{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{2 u}{u ^{2} + 4} d u + \int \frac{3}{u ^{2} + 4} d u

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

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

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

u = x + 1

代回

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

化简

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

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

解答补常数

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

作图

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