What is the integral from 0 to 8/9 of 1/(81x^2+64) ?

The integral from 0 to 8/9 of 1/(81x^2+64) is pi/(288)

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integral from 0 to 8/9 of 1/(81x^2+64)

\int_{0}^{\frac{8}{9}} \frac{1}{81 x ^{2} + 64} d x

\frac{π}{288}

Decimal

0.01090 \dots

Solution steps

\int_{0}^{\frac{8}{9}} \frac{1}{81 x ^{2} + 64} d x

Apply Integral Substitution

= \int_{0}^{1} \frac{1}{72 ( 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}{72} \cdot \int_{0}^{1} \frac{1}{u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= \frac{1}{72} [arctan (u)]_{0}^{1}

Compute the boundaries:

\frac{π}{4}

= \frac{1}{72} \cdot \frac{π}{4}

Simplify

= \frac{π}{288}

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What is the integral from 0 to 8/9 of 1/(81x^2+64) ?
The integral from 0 to 8/9 of 1/(81x^2+64) is pi/(288)