What is the integral of (x+3)/(sqrt(x^2+6x)) ?

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

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

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

x^{2} + 6 x + C

Solution steps

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

Expand

\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

Apply the Sum Rule:

\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)

Simplify

= x^{2} + 6 x

Add a constant to the solution

= x^{2} + 6 x + C

What is the integral of (x+3)/(sqrt(x^2+6x)) ?
The integral of (x+3)/(sqrt(x^2+6x)) is sqrt(x^2+6x)+C