What is the integral from-1/3 to 1/3 of 1/(1-x^2) ?

The integral from-1/3 to 1/3 of 1/(1-x^2) is ln(2)

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integral from-1/3 to 1/3 of 1/(1-x^2)

\int_{- \frac{1}{3}}^{\frac{1}{3}} \frac{1}{1 - x ^{2}} d x

ln (2)

Decimal

0.69314 \dots

Solution steps

\int_{- \frac{1}{3}}^{\frac{1}{3}} \frac{1}{1 - x ^{2}} d x

Use the common integral:

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

= [\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{- \frac{1}{3}}^{\frac{1}{3}}

Simplify

[\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{- \frac{1}{3}}^{\frac{1}{3}} : [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{- \frac{1}{3}}^{\frac{1}{3}}

= [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{- \frac{1}{3}}^{\frac{1}{3}}

Compute the boundaries:

\frac{- 2 ln ( \frac{2}{3} ) + 2 ln ( \frac{4}{3} )}{2}

= \frac{- 2 ln ( \frac{2}{3} ) + 2 ln ( \frac{4}{3} )}{2}

Simplify

= ln (2)

What is the integral from-1/3 to 1/3 of 1/(1-x^2) ?
The integral from-1/3 to 1/3 of 1/(1-x^2) is ln(2)