What is the integral of (x+5)/(x^2+2x-15) ?

The integral of (x+5)/(x^2+2x-15) is ln|x-3|+C

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integral of (x+5)/(x^2+2x-15)

\int \frac{x + 5}{x ^{2} + 2 x - 15} d x

ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{x + 5}{x ^{2} + 2 x - 15} d x

Complete the square

x^{2} + 2 x - 15 : (x + 1)^{2} - 16

= \int \frac{x + 5}{( x + 1 ) ^{2} - 16} d x

Apply u-substitution

= \int \frac{1}{u - 4} d u

Apply u-substitution

= \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= ln ∣ v ∣

Substitute back

= ln ∣ x + 1 - 4 ∣

Simplify

= ln ∣ x - 3 ∣

Add a constant to the solution

= ln ∣ x - 3 ∣ + C

What is the integral of (x+5)/(x^2+2x-15) ?
The integral of (x+5)/(x^2+2x-15) is ln|x-3|+C