What is the integral from 0 to 5/2 of 1/(4x^2+25) ?

The integral from 0 to 5/2 of 1/(4x^2+25) is pi/(40)

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integral from 0 to 5/2 of 1/(4x^2+25)

\int_{0}^{\frac{5}{2}} \frac{1}{4 x ^{2} + 25} d x

\frac{π}{40}

Decimal

0.07853 \dots

Solution steps

\int_{0}^{\frac{5}{2}} \frac{1}{4 x ^{2} + 25} d x

Apply Integral Substitution

= \int_{0}^{1} \frac{1}{10 ( 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}{10} \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}{10} [arctan (u)]_{0}^{1}

Compute the boundaries:

\frac{π}{4}

= \frac{1}{10} \cdot \frac{π}{4}

Simplify

= \frac{π}{40}

\int_{0}^{1} 8 x^{2} + 1 d x

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\int_{- 4}^{- 2} (x^{2} + \frac{1}{x ^{3}}) d x

\int_{0}^{1} π (1 - x)^{2} d x

What is the integral from 0 to 5/2 of 1/(4x^2+25) ?
The integral from 0 to 5/2 of 1/(4x^2+25) is pi/(40)