What is the integral of 3/(x^2-x-2) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 3/(x^2-x-2)

\int \frac{3}{x ^{2} - x - 2} d x

- ln \frac{2}{3} + \frac{2}{3} x + ln - \frac{4}{3} + \frac{2}{3} x + C

Solution steps

\int \frac{3}{x ^{2} - x - 2} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{x ^{2} - x - 2} d x

Complete the square

x^{2} - x - 2 : (x - \frac{1}{2})^{2} - \frac{9}{4}

= 3 \cdot \int \frac{1}{( x - \frac{1}{2} ) ^{2} - \frac{9}{4}} d x

Apply u-substitution

: \int \frac{4}{4 u ^{2} - 9} d u

= 3 \cdot \int \frac{4}{4 u ^{2} - 9} d u

Take the constant out:

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

= 3 \cdot 4 \cdot \int \frac{1}{4 u ^{2} - 9} d u

Apply Integral Substitution

= 3 \cdot 4 \cdot \int \frac{1}{6 ( v ^{2} - 1 )} d v

Take the constant out:

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

= 3 \cdot 4 \cdot \frac{1}{6} \cdot \int \frac{1}{v ^{2} - 1} d v

Factor

v^{2} - 1 : - (- v^{2} + 1)

= 3 \cdot 4 \cdot \frac{1}{6} \cdot \int \frac{1}{- ( - v ^{2} + 1 )} d v

Take the constant out:

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

= 3 \cdot 4 \cdot \frac{1}{6} (- \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}

= 3 \cdot 4 \cdot \frac{1}{6} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Substitute back

= 3 \cdot 4 \cdot \frac{1}{6} (- (\frac{ln \frac{2}{3} ( x - \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{3} ( x - \frac{1}{2} ) - 1}{2}))

Simplify

3 \cdot 4 \cdot \frac{1}{6} (- (\frac{ln \frac{2}{3} ( x - \frac{1}{2} ) + 1}{2} - \frac{ln \frac{2}{3} ( x - \frac{1}{2} ) - 1}{2})) : - ln \frac{2}{3} + \frac{2}{3} x + ln - \frac{4}{3} + \frac{2}{3} x

= - ln \frac{2}{3} + \frac{2}{3} x + ln - \frac{4}{3} + \frac{2}{3} x

Add a constant to the solution

= - ln \frac{2}{3} + \frac{2}{3} x + ln - \frac{4}{3} + \frac{2}{3} x + C

\int ln (3 x + 4) d x

\frac{d y}{d x} = \frac{y ^{2}}{x - 1}

\frac{\partial}{\partial y} (x^{2} - 3 x y + y^{2})

f (x) = 2 x^{2} - x

\int \frac{1}{t ^{2} - 2 t + 26} d t

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