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

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

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

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

ln (3)

Decimal Notation

1.09861 \dots

Solution steps

\int_{- \frac{1}{2}}^{\frac{1}{2}} \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}{2}}^{\frac{1}{2}}

Simplify

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

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

Compute the boundaries:

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

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

Simplify

= ln (3)

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