What is the integral of (3x)/(x^2-x+6) ?

The integral of (3x)/(x^2-x+6) is 6(1/4 ln|4x^2-4x+24|+1/(2sqrt(23))arctan((2x-1)/(sqrt(23))))+C

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integral of (3x)/(x^2-x+6)

\int \frac{3 x}{x ^{2} - x + 6} d x

6 (\frac{1}{4} ln 4 x^{2} - 4 x + 24 + \frac{1}{2 23} arctan (\frac{2 x - 1}{23})) + C

Solution steps

\int \frac{3 x}{x ^{2} - x + 6} d x

Take the constant out:

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

= 3 \cdot \int \frac{x}{x ^{2} - x + 6} d x

Complete the square

x^{2} - x + 6 : (x - \frac{1}{2})^{2} + \frac{23}{4}

= 3 \cdot \int \frac{x}{( x - \frac{1}{2} ) ^{2} + \frac{23}{4}} d x

Apply u-substitution

: \int \frac{2 ( 2 u + 1 )}{4 u ^{2} + 23} d u

= 3 \cdot \int \frac{2 ( 2 u + 1 )}{4 u ^{2} + 23} d u

Take the constant out:

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

= 3 \cdot 2 \cdot \int \frac{2 u + 1}{4 u ^{2} + 23} d u

Expand

\frac{2 u + 1}{4 u ^{2} + 23} : \frac{2 u}{4 u ^{2} + 23} + \frac{1}{4 u ^{2} + 23}

Apply the Sum Rule:

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

= 3 \cdot 2 (\int \frac{2 u}{4 u ^{2} + 23} d u + \int \frac{1}{4 u ^{2} + 23} d u)

\int \frac{2 u}{4 u ^{2} + 23} d u = \frac{1}{4} ln 4 u^{2} + 23

\int \frac{1}{4 u ^{2} + 23} d u = \frac{1}{2 23} arctan (\frac{2}{23} u)

= 3 \cdot 2 (\frac{1}{4} ln 4 u^{2} + 23 + \frac{1}{2 23} arctan (\frac{2}{23} u))

Substitute back

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

= 3 \cdot 2 (\frac{1}{4} ln 4 (x - \frac{1}{2})^{2} + 23 + \frac{1}{2 23} arctan (\frac{2}{23} (x - \frac{1}{2})))

Simplify

3 \cdot 2 (\frac{1}{4} ln 4 (x - \frac{1}{2})^{2} + 23 + \frac{1}{2 23} arctan (\frac{2}{23} (x - \frac{1}{2}))) : 6 (\frac{1}{4} ln 4 x^{2} - 4 x + 24 + \frac{1}{2 23} arctan (\frac{2 x - 1}{23}))

= 6 (\frac{1}{4} ln 4 x^{2} - 4 x + 24 + \frac{1}{2 23} arctan (\frac{2 x - 1}{23}))

Add a constant to the solution

= 6 (\frac{1}{4} ln 4 x^{2} - 4 x + 24 + \frac{1}{2 23} arctan (\frac{2 x - 1}{23})) + C

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What is the integral of (3x)/(x^2-x+6) ?
The integral of (3x)/(x^2-x+6) is 6(1/4 ln|4x^2-4x+24|+1/(2sqrt(23))arctan((2x-1)/(sqrt(23))))+C