integral of (x+3)/(sqrt(x^2+6x))

解答

\int \frac{x + 3}{x ^{2} + 6 x} d x

x^{2} + 6 x + C

求解步骤

\int \frac{x + 3}{x ^{2} + 6 x} d x

乘开

\frac{x + 3}{x ^{2} + 6 x} : \frac{x}{x ^{2} + 6 x} + \frac{3}{x ^{2} + 6 x}

= \int \frac{x}{x ^{2} + 6 x} + \frac{3}{x ^{2} + 6 x} d x

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{x}{x ^{2} + 6 x} d x + \int \frac{3}{x ^{2} + 6 x} d x

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

\int \frac{3}{x ^{2} + 6 x} d x = 3 ln (\frac{1}{3} x^{2} + 6 x + x + 3)

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

化简

= x^{2} + 6 x

解答补常数

= x^{2} + 6 x + C

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