What is the integral of 1/((5-4x-x^2)) ?

The integral of 1/((5-4x-x^2)) is 1/6 (ln|(x+2)/3+1|-ln|(x+2)/3-1|)+C

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integral of 1/((5-4x-x^2))

\int \frac{1}{( 5 - 4 x - x ^{2} )} d x

\frac{1}{6} (ln \frac{x + 2}{3} + 1 - ln \frac{x + 2}{3} - 1) + C

Solution steps

\int \frac{1}{5 - 4 x - x ^{2}} d x

Complete the square

5 - 4 x - x^{2} : - (x + 2)^{2} + 9

= \int \frac{1}{- ( x + 2 ) ^{2} + 9} d x

Apply u-substitution

= \int \frac{1}{- u ^{2} + 9} 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 + 2}{3} + 1}{2} - \frac{ln \frac{x + 2}{3} - 1}{2})

Simplify

\frac{1}{3} (\frac{ln \frac{x + 2}{3} + 1}{2} - \frac{ln \frac{x + 2}{3} - 1}{2}) : \frac{1}{6} (ln \frac{x + 2}{3} + 1 - ln \frac{x + 2}{3} - 1)

= \frac{1}{6} (ln \frac{x + 2}{3} + 1 - ln \frac{x + 2}{3} - 1)

Add a constant to the solution

= \frac{1}{6} (ln \frac{x + 2}{3} + 1 - ln \frac{x + 2}{3} - 1) + C

What is the integral of 1/((5-4x-x^2)) ?
The integral of 1/((5-4x-x^2)) is 1/6 (ln|(x+2)/3+1|-ln|(x+2)/3-1|)+C