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

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

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

\int \frac{( x + 1 )}{x ^{2} + x + 1} d x

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

Solution steps

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

Expand

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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