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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Complete the square

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

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

Apply u-substitution

: \int \frac{2 ( 2 u - 1 )}{4 u ^{2} + 3} d u

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot 2 \cdot \int \frac{2 u - 1}{4 u ^{2} + 3} d u

Expand

\frac{2 u - 1}{4 u ^{2} + 3} : \frac{2 u}{4 u ^{2} + 3} - \frac{1}{4 u ^{2} + 3}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{2 u}{4 u ^{2} + 3} d u = \frac{1}{4} ln 4 u^{2} + 3

\int \frac{1}{4 u ^{2} + 3} d u = \frac{1}{2 3} arctan (\frac{2}{3} u)

= 2 \cdot 2 (\frac{1}{4} ln 4 u^{2} + 3 - \frac{1}{2 3} arctan (\frac{2}{3} u))

Substitute back

u = x + \frac{1}{2}

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

Simplify

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

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

Add a constant to the solution

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

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