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

The integral from 0 to 1 of 1/(4+4x^2) is pi/(16)

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

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

\frac{π}{16}

Decimal

0.19634 \dots

Solution steps

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

\frac{1}{4 + 4 x ^{2}} = \frac{1}{4} \frac{1}{1 + x ^{2}}

= \int_{0}^{1} \frac{1}{4} \cdot \frac{1}{1 + x ^{2}} d x

Take the constant out:

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

= \frac{1}{4} \cdot \int_{0}^{1} \frac{1}{1 + x ^{2}} d x

Use the common integral:

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

= \frac{1}{4} [arctan (x)]_{0}^{1}

Compute the boundaries:

\frac{π}{4}

= \frac{1}{4} \cdot \frac{π}{4}

Simplify

= \frac{π}{16}

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What is the integral from 0 to 1 of 1/(4+4x^2) ?
The integral from 0 to 1 of 1/(4+4x^2) is pi/(16)