What is the integral from 0 to 3/2 of 1/(9-x^2) ?

The integral from 0 to 3/2 of 1/(9-x^2) is (ln(3))/6

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integral from 0 to 3/2 of 1/(9-x^2)

\int_{0}^{\frac{3}{2}} \frac{1}{9 - x ^{2}} d x

\frac{ln ( 3 )}{6}

Solution steps

\int_{0}^{\frac{3}{2}} \frac{1}{9 - x ^{2}} d x

Apply Integral Substitution

= \int_{0}^{\frac{1}{2}} \frac{1}{3 ( - 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}{3} \cdot \int_{0}^{\frac{1}{2}} \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}{3} [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{2}}

Simplify

\frac{1}{3} [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{2}} : \frac{1}{3} [\frac{ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{2}}

= \frac{1}{3} [\frac{ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{2}}

Compute the boundaries:

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

= \frac{1}{3} \cdot \frac{ln ( 2 ) + ln ( \frac{3}{2} )}{2}

Simplify

= \frac{ln ( 3 )}{6}

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What is the integral from 0 to 3/2 of 1/(9-x^2) ?
The integral from 0 to 3/2 of 1/(9-x^2) is (ln(3))/6