What is the integral of (x+5)/(sqrt(x^2+6x+13)) ?

The integral of (x+5)/(sqrt(x^2+6x+13)) is sqrt(x^2+6x+13)+2ln(1/2 |x+3+sqrt(x^2+6x+13)|)+C

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

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

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

Solution steps

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

Expand

\frac{x + 5}{x ^{2} + 6 x + 13} : \frac{x}{x ^{2} + 6 x + 13} + \frac{5}{x ^{2} + 6 x + 13}

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

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

\int \frac{5}{x ^{2} + 6 x + 13} d x = 5 ln (\frac{1}{2} x + 3 + x^{2} + 6 x + 13)

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (x+5)/(sqrt(x^2+6x+13)) ?
The integral of (x+5)/(sqrt(x^2+6x+13)) is sqrt(x^2+6x+13)+2ln(1/2 |x+3+sqrt(x^2+6x+13)|)+C