What is the integral from 0 to 1 of (24)/(x^2-36) ?

The integral from 0 to 1 of (24)/(x^2-36) is -2(ln(7/6)-ln(5/6))

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integral from 0 to 1 of (24)/(x^2-36)

\int_{0}^{1} \frac{24}{x ^{2} - 36} d x

- 2 (ln (\frac{7}{6}) - ln (\frac{5}{6}))

Decimal

- 0.67294 \dots

Solution steps

\int_{0}^{1} \frac{24}{x ^{2} - 36} d x

Take the constant out:

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

= 24 \cdot \int_{0}^{1} \frac{1}{x ^{2} - 36} d x

Apply Integral Substitution

= 24 \cdot \int_{0}^{\frac{1}{6}} \frac{1}{6 ( u ^{2} - 1 )} d u

Take the constant out:

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

= 24 \cdot \frac{1}{6} \cdot \int_{0}^{\frac{1}{6}} \frac{1}{u ^{2} - 1} d u

Factor

u^{2} - 1 : - (- u^{2} + 1)

= 24 \cdot \frac{1}{6} \cdot \int_{0}^{\frac{1}{6}} \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

= 24 \cdot \frac{1}{6} (- \int_{0}^{\frac{1}{6}} \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}

= 24 \cdot \frac{1}{6} (- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{6}})

Simplify

24 \cdot \frac{1}{6} (- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{0}^{\frac{1}{6}}) : - 4 [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{0}^{\frac{1}{6}}

= - 4 [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{0}^{\frac{1}{6}}

Compute the boundaries:

\frac{ln ( \frac{7}{6} ) - ln ( \frac{5}{6} )}{2}

= - 4 \cdot \frac{ln ( \frac{7}{6} ) - ln ( \frac{5}{6} )}{2}

Simplify

= - 2 (ln (\frac{7}{6}) - ln (\frac{5}{6}))

What is the integral from 0 to 1 of (24)/(x^2-36) ?
The integral from 0 to 1 of (24)/(x^2-36) is -2(ln(7/6)-ln(5/6))