What is the integral from 0 to 1/2 of 2/(1-x^2) ?

The integral from 0 to 1/2 of 2/(1-x^2) is ln(3)

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integral from 0 to 1/2 of 2/(1-x^2)

\int_{0}^{\frac{1}{2}} \frac{2}{1 - x ^{2}} d x

ln (3)

Decimal

1.09861 \dots

Solution steps

\int_{0}^{\frac{1}{2}} \frac{2}{1 - x ^{2}} d x

Take the constant out:

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

= 2 \cdot \int_{0}^{\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}

= 2 [\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{0}^{\frac{1}{2}}

Simplify

2 [\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{0}^{\frac{1}{2}} : 2 [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{0}^{\frac{1}{2}}

= 2 [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{0}^{\frac{1}{2}}

Compute the boundaries:

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

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

Simplify

= ln (3)

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What is the integral from 0 to 1/2 of 2/(1-x^2) ?
The integral from 0 to 1/2 of 2/(1-x^2) is ln(3)