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

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

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

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

- ln 1 - 2 x - x^{2} + C

Solution steps

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

Apply u-substitution

: \int - \frac{1}{2 u} d u

= 2 \cdot \int - \frac{1}{2 u} d u

Take the constant out:

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

= 2 (- \frac{1}{2} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 (- \frac{1}{2} ln ∣ u ∣)

Substitute back

u = 1 - 2 x - x^{2}

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

Simplify

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

= - ln 1 - 2 x - x^{2}

Add a constant to the solution

= - ln 1 - 2 x - x^{2} + C

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