What is the integral of (2x+1)/(4x^2+12x-7) ?

The integral of (2x+1)/(4x^2+12x-7) is 1/8 ln|2x-1|+3/8 ln|2x+7|+C

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integral of (2x+1)/(4x^2+12x-7)

\int \frac{2 x + 1}{4 x ^{2} + 12 x - 7} d x

\frac{1}{8} ln ∣ 2 x - 1 ∣ + \frac{3}{8} ln ∣ 2 x + 7 ∣ + C

Solution steps

\int \frac{2 x + 1}{4 x ^{2} + 12 x - 7} d x

Take the partial fraction of

\frac{2 x + 1}{4 x ^{2} + 12 x - 7} : \frac{1}{4 ( 2 x - 1 )} + \frac{3}{4 ( 2 x + 7 )}

= \int \frac{1}{4 ( 2 x - 1 )} + \frac{3}{4 ( 2 x + 7 )} 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{1}{4 ( 2 x - 1 )} d x + \int \frac{3}{4 ( 2 x + 7 )} d x

\int \frac{1}{4 ( 2 x - 1 )} d x = \frac{1}{8} ln ∣ 2 x - 1 ∣

\int \frac{3}{4 ( 2 x + 7 )} d x = \frac{3}{8} ln ∣ 2 x + 7 ∣

= \frac{1}{8} ln ∣ 2 x - 1 ∣ + \frac{3}{8} ln ∣ 2 x + 7 ∣

Add a constant to the solution

= \frac{1}{8} ln ∣ 2 x - 1 ∣ + \frac{3}{8} ln ∣ 2 x + 7 ∣ + C

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What is the integral of (2x+1)/(4x^2+12x-7) ?
The integral of (2x+1)/(4x^2+12x-7) is 1/8 ln|2x-1|+3/8 ln|2x+7|+C