What is the integral of (2x+3)/(sqrt(x^2+2x+5)) ?

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

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integral of (2x+3)/(sqrt(x^2+2x+5))

\int \frac{2 x + 3}{x ^{2} + 2 x + 5} d x

2 x^{2} + 2 x + 5 + ln (\frac{1}{2} x + 1 + x^{2} + 2 x + 5) + C

Solution steps

\int \frac{2 x + 3}{x ^{2} + 2 x + 5} d x

Expand

\frac{2 x + 3}{x ^{2} + 2 x + 5} : \frac{2 x}{x ^{2} + 2 x + 5} + \frac{3}{x ^{2} + 2 x + 5}

= \int \frac{2 x}{x ^{2} + 2 x + 5} + \frac{3}{x ^{2} + 2 x + 5} 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} + 2 x + 5} d x + \int \frac{3}{x ^{2} + 2 x + 5} d x

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

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

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

Simplify

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

= 2 x^{2} + 2 x + 5 + ln (\frac{1}{2} x + 1 + x^{2} + 2 x + 5)

Add a constant to the solution

= 2 x^{2} + 2 x + 5 + ln (\frac{1}{2} x + 1 + x^{2} + 2 x + 5) + C

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