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受欢迎的微积分 >

integral of (5x+1)/(x^2-6x+13)

解答

\int \frac{5 x + 1}{x ^{2} - 6 x + 13} d x

解答

\frac{5}{2} ln x^{2} - 6 x + 13 + 8 arctan (\frac{1}{2} (x - 3)) + C

求解步骤

\int \frac{5 x + 1}{x ^{2} - 6 x + 13} d x

对

x^{2} - 6 x + 13

配方:

(x - 3)^{2} + 4

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

使用换元积分法

= \int \frac{5 u + 16}{u ^{2} + 4} d u

乘开

\frac{5 u + 16}{u ^{2} + 4} : \frac{5 u}{u ^{2} + 4} + \frac{16}{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{5 u}{u ^{2} + 4} d u + \int \frac{16}{u ^{2} + 4} d u

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

\int \frac{16}{u ^{2} + 4} d u = 8 arctan (\frac{u}{2})

= \frac{5}{2} ln u^{2} + 4 + 8 arctan (\frac{u}{2})

u = x - 3

代回

= \frac{5}{2} ln (x - 3)^{2} + 4 + 8 arctan (\frac{x - 3}{2})

化简

\frac{5}{2} ln (x - 3)^{2} + 4 + 8 arctan (\frac{x - 3}{2}) : \frac{5}{2} ln x^{2} - 6 x + 13 + 8 arctan (\frac{1}{2} (x - 3))

= \frac{5}{2} ln x^{2} - 6 x + 13 + 8 arctan (\frac{1}{2} (x - 3))

解答补常数

= \frac{5}{2} ln x^{2} - 6 x + 13 + 8 arctan (\frac{1}{2} (x - 3)) + C

作图

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