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受欢迎的微积分 >

integral of x/(x^2+6x+13)

解答

\int \frac{x}{x ^{2} + 6 x + 13} d x

解答

\frac{1}{2} ln x^{2} + 6 x + 13 - \frac{3}{2} arctan (\frac{1}{2} (x + 3)) + C

求解步骤

\int \frac{x}{x ^{2} + 6 x + 13} d x

对

x^{2} + 6 x + 13

配方:

(x + 3)^{2} + 4

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

使用换元积分法

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

乘开

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

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

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

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

u = x + 3

代回

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

化简

\frac{1}{2} ln (x + 3)^{2} + 4 - \frac{3}{2} arctan (\frac{x + 3}{2}) : \frac{1}{2} ln x^{2} + 6 x + 13 - \frac{3}{2} arctan (\frac{1}{2} (x + 3))

= \frac{1}{2} ln x^{2} + 6 x + 13 - \frac{3}{2} arctan (\frac{1}{2} (x + 3))

解答补常数

= \frac{1}{2} ln x^{2} + 6 x + 13 - \frac{3}{2} arctan (\frac{1}{2} (x + 3)) + C

作图

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