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

The integral of 2/(x^2+2x-15) is -1/4 (ln|(x+1)/4+1|-ln|(x+1)/4-1|)+C

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

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

- \frac{1}{4} (ln \frac{x + 1}{4} + 1 - ln \frac{x + 1}{4} - 1) + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{1}{x ^{2} + 2 x - 15} d x

Factor

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

= 2 \cdot \int \frac{1}{- ( - x ^{2} - 2 x + 15 )} d x

Take the constant out:

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

= 2 (- \int \frac{1}{- x ^{2} - 2 x + 15} d x)

Complete the square

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

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

Apply u-substitution

: \int \frac{1}{- u ^{2} + 16} d u

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

Apply Integral Substitution

= 2 (- \int \frac{1}{4 ( - v ^{2} + 1 )} d v)

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= 2 (- \frac{1}{4} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

= 2 (- \frac{1}{4} (\frac{ln \frac{x + 1}{4} + 1}{2} - \frac{ln \frac{x + 1}{4} - 1}{2}))

Simplify

2 (- \frac{1}{4} (\frac{ln \frac{x + 1}{4} + 1}{2} - \frac{ln \frac{x + 1}{4} - 1}{2})) : - \frac{1}{4} (ln \frac{x + 1}{4} + 1 - ln \frac{x + 1}{4} - 1)

= - \frac{1}{4} (ln \frac{x + 1}{4} + 1 - ln \frac{x + 1}{4} - 1)

Add a constant to the solution

= - \frac{1}{4} (ln \frac{x + 1}{4} + 1 - ln \frac{x + 1}{4} - 1) + C

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What is the integral of 2/(x^2+2x-15) ?
The integral of 2/(x^2+2x-15) is -1/4 (ln|(x+1)/4+1|-ln|(x+1)/4-1|)+C