What is the integral of (2x+5)/(sqrt(x^2+4x+13)) ?

The integral of (2x+5)/(sqrt(x^2+4x+13)) is 2sqrt(x^2+4x+13)+ln(1/3 |x+2+sqrt(x^2+4x+13)|)+C

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integral of (2x+5)/(sqrt(x^2+4x+13))

\int \frac{2 x + 5}{x ^{2} + 4 x + 13} d x

2 x^{2} + 4 x + 13 + ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13) + C

Solution steps

\int \frac{2 x + 5}{x ^{2} + 4 x + 13} d x

Expand

\frac{2 x + 5}{x ^{2} + 4 x + 13} : \frac{2 x}{x ^{2} + 4 x + 13} + \frac{5}{x ^{2} + 4 x + 13}

= \int \frac{2 x}{x ^{2} + 4 x + 13} + \frac{5}{x ^{2} + 4 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{2 x}{x ^{2} + 4 x + 13} d x + \int \frac{5}{x ^{2} + 4 x + 13} d x

\int \frac{2 x}{x ^{2} + 4 x + 13} d x = 2 (x^{2} + 4 x + 13 - 2 ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13))

\int \frac{5}{x ^{2} + 4 x + 13} d x = 5 ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13)

= 2 (x^{2} + 4 x + 13 - 2 ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13)) + 5 ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13)

Simplify

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

= 2 x^{2} + 4 x + 13 + ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13)

Add a constant to the solution

= 2 x^{2} + 4 x + 13 + ln (\frac{1}{3} x + 2 + x^{2} + 4 x + 13) + C

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What is the integral of (2x+5)/(sqrt(x^2+4x+13)) ?
The integral of (2x+5)/(sqrt(x^2+4x+13)) is 2sqrt(x^2+4x+13)+ln(1/3 |x+2+sqrt(x^2+4x+13)|)+C