What is the integral of (4x+2)/(sqrt(x^2+5x-1)) ?

The integral of (4x+2)/(sqrt(x^2+5x-1)) is

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integral of (4x+2)/(sqrt(x^2+5x-1))

\int \frac{4 x + 2}{x ^{2} + 5 x - 1} d x

4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{1}{29} (2 x + 5) + C

Solution steps

\int \frac{4 x + 2}{x ^{2} + 5 x - 1} d x

Expand

\frac{4 x + 2}{x ^{2} + 5 x - 1} : \frac{4 x}{x ^{2} + 5 x - 1} + \frac{2}{x ^{2} + 5 x - 1}

= \int \frac{4 x}{x ^{2} + 5 x - 1} + \frac{2}{x ^{2} + 5 x - 1} 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{4 x}{x ^{2} + 5 x - 1} d x + \int \frac{2}{x ^{2} + 5 x - 1} d x

\int \frac{4 x}{x ^{2} + 5 x - 1} d x = 4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{4 x ^{2} + 20 x - 4}{29} + \frac{2}{29} x + \frac{5}{29})

\int \frac{2}{x ^{2} + 5 x - 1} d x = 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}

= 4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{4 x ^{2} + 20 x - 4}{29} + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}

Simplify

4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{4 x ^{2} + 20 x - 4}{29} + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29} : 4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{1}{29} (2 x + 5)

= 4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{1}{29} (2 x + 5)

Add a constant to the solution

= 4 (x^{2} + 5 x - 1 - \frac{5}{2} ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{2}{29} x + \frac{5}{29}) + 2 ln \frac{1}{29} (4 x^{2} + 20 x - 4) + \frac{1}{29} (2 x + 5) + C

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What is the integral of (4x+2)/(sqrt(x^2+5x-1)) ?
The integral of (4x+2)/(sqrt(x^2+5x-1)) is