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

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

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

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

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

Solution steps

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

Simplify

= \int \frac{1}{2 x ^{2} - 1} 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 = 2 x

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

Simplify

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

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

Add a constant to the solution

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

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