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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

integral

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

- \frac{1}{4} (ln \frac{x}{2} + 1 - ln \frac{x}{2} - 1) + C

Solution steps

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

Apply Integral Substitution

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

Take the constant out:

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

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

Factor

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

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

u = \frac{x}{2}

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

Simplify

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

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

Add a constant to the solution

= - \frac{1}{4} (ln \frac{x}{2} + 1 - ln \frac{x}{2} - 1) + C

\frac{4 x ^{4}}{( 1 + x ^{2} )}

\frac{4 t}{3 t ^{2} - 7}

u^{\frac{1}{2}}

\frac{1}{( t ^{2} + t + 1 )}

x^{2} e^{\frac{x ^{2}}{2}}

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