What is the integral of ((2x+1))/(x^2+4x-21) ?

The integral of ((2x+1))/(x^2+4x-21) is 7/10 ln|x-3|+13/10 ln|x+7|+C

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integral of ((2x+1))/(x^2+4x-21)

\int \frac{( 2 x + 1 )}{x ^{2} + 4 x - 21} d x

\frac{7}{10} ln ∣ x - 3 ∣ + \frac{13}{10} ln ∣ x + 7 ∣ + C

Solution steps

\int \frac{2 x + 1}{x ^{2} + 4 x - 21} d x

Expand

\frac{2 x + 1}{x ^{2} + 4 x - 21} : \frac{2 x}{x ^{2} + 4 x - 21} + \frac{1}{x ^{2} + 4 x - 21}

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{2 x}{x ^{2} + 4 x - 21} d x + \int \frac{1}{x ^{2} + 4 x - 21} d x

\int \frac{2 x}{x ^{2} + 4 x - 21} d x = 2 (\frac{3}{10} ln ∣ x - 3 ∣ + \frac{7}{10} ln ∣ x + 7 ∣)

\int \frac{1}{x ^{2} + 4 x - 21} d x = \frac{1}{10} ln ∣ x - 3 ∣ - \frac{1}{10} ln ∣ x + 7 ∣

= 2 (\frac{3}{10} ln ∣ x - 3 ∣ + \frac{7}{10} ln ∣ x + 7 ∣) + \frac{1}{10} ln ∣ x - 3 ∣ - \frac{1}{10} ln ∣ x + 7 ∣

Simplify

2 (\frac{3}{10} ln ∣ x - 3 ∣ + \frac{7}{10} ln ∣ x + 7 ∣) + \frac{1}{10} ln ∣ x - 3 ∣ - \frac{1}{10} ln ∣ x + 7 ∣ : \frac{7}{10} ln ∣ x - 3 ∣ + \frac{13}{10} ln ∣ x + 7 ∣

= \frac{7}{10} ln ∣ x - 3 ∣ + \frac{13}{10} ln ∣ x + 7 ∣

Add a constant to the solution

= \frac{7}{10} ln ∣ x - 3 ∣ + \frac{13}{10} ln ∣ x + 7 ∣ + C

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What is the integral of ((2x+1))/(x^2+4x-21) ?
The integral of ((2x+1))/(x^2+4x-21) is 7/10 ln|x-3|+13/10 ln|x+7|+C