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

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

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integral of (3x+5)/(sqrt(x^2+x+1))

\int \frac{3 x + 5}{x ^{2} + x + 1} d x

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

Solution steps

\int \frac{3 x + 5}{x ^{2} + x + 1} d x

Expand

\frac{3 x + 5}{x ^{2} + x + 1} : \frac{3 x}{x ^{2} + x + 1} + \frac{5}{x ^{2} + x + 1}

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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