What is the integral of (6x-3)/(x^2-x+7) ?

The integral of (6x-3)/(x^2-x+7) is 3ln|36x^2-36x+252|+C

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integral of (6x-3)/(x^2-x+7)

\int \frac{6 x - 3}{x ^{2} - x + 7} d x

3 ln 36 x^{2} - 36 x + 252 + C

Solution steps

\int \frac{6 x - 3}{x ^{2} - x + 7} d x

Apply u-substitution

: \int \frac{6 u}{u ^{2} + 243} d u

= \int \frac{6 u}{u ^{2} + 243} d u

Take the constant out:

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

= 6 \cdot \int \frac{u}{u ^{2} + 243} d u

Apply u-substitution

: \int \frac{1}{2 v} d v

= 6 \cdot \int \frac{1}{2 v} d v

Take the constant out:

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

= 6 \cdot \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 6 \cdot \frac{1}{2} ln ∣ v ∣

Substitute back

= 6 \cdot \frac{1}{2} ln (6 x - 3)^{2} + 243

Simplify

6 \cdot \frac{1}{2} ln (6 x - 3)^{2} + 243 : 3 ln 36 x^{2} - 36 x + 252

= 3 ln 36 x^{2} - 36 x + 252

Add a constant to the solution

= 3 ln 36 x^{2} - 36 x + 252 + C

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What is the integral of (6x-3)/(x^2-x+7) ?
The integral of (6x-3)/(x^2-x+7) is 3ln|36x^2-36x+252|+C