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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

\frac{d}{d x} (- 3 x)

\int 10 - e^{2 x} d x

\frac{d}{d x} (2 + 2 + x)

\frac{d P}{d t} = Pt, P (1) = 3

\int - x^{2} + 9 x d x

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