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

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

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

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

- 2 ln 2 x + 1 + 2 ln 2 x - 1 + C

Solution steps

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

Take the constant out:

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

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

Apply Integral Substitution

= - 4 \cdot \int \frac{1}{2 ( 1 - u ^{2} )} d u

Take the constant out:

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

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

Use the common integral:

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

= - 4 \cdot \frac{1}{2} (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

Substitute back

u = 2 x

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

Simplify

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

= - 2 ln 2 x + 1 + 2 ln 2 x - 1

Add a constant to the solution

= - 2 ln 2 x + 1 + 2 ln 2 x - 1 + C

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