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

The integral of 1/(-x^2+2x+2) is 1/(2sqrt(3))(ln|(x-1)/(sqrt(3))+1|-ln|(x-1)/(sqrt(3))-1|)+C

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

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

\frac{1}{2 3} (ln \frac{x - 1}{3} + 1 - ln \frac{x - 1}{3} - 1) + C

Solution steps

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

Complete the square

- x^{2} + 2 x + 2 : - (x - 1)^{2} + 3

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

Apply u-substitution

: \int \frac{1}{- u ^{2} + 3} d u

= \int \frac{1}{- u ^{2} + 3} d u

Apply Integral Substitution

= \int \frac{1}{3 ( - v ^{2} + 1 )} d v

Take the constant out:

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

= \frac{1}{3} \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}

= \frac{1}{3} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Substitute back

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

Simplify

\frac{1}{3} \frac{ln \frac{x - 1}{3} + 1}{2} - \frac{ln \frac{x - 1}{3} - 1}{2} : \frac{1}{2 3} (ln \frac{x - 1}{3} + 1 - ln \frac{x - 1}{3} - 1)

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

Add a constant to the solution

= \frac{1}{2 3} (ln \frac{x - 1}{3} + 1 - ln \frac{x - 1}{3} - 1) + C

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