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

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

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

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

\frac{1}{4} (arctan (π - \frac{1}{2}) + arctan (\frac{1}{2}))

Decimal Notation

0.41813 \dots

Solution steps

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

Complete the square

4 x^{2} - 4 x + 5 : 4 (x - \frac{1}{2})^{2} + 4

= \int_{0}^{π} \frac{1}{4 ( x - \frac{1}{2} ) ^{2} + 4} d x

Apply u-substitution

: \int_{- \frac{1}{2}}^{π - \frac{1}{2}} \frac{1}{4 u ^{2} + 4} d u

= \int_{- \frac{1}{2}}^{π - \frac{1}{2}} \frac{1}{4 u ^{2} + 4} d u

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

= \int_{- \frac{1}{2}}^{π - \frac{1}{2}} \frac{1}{4} \cdot \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

= \frac{1}{4} \cdot \int_{- \frac{1}{2}}^{π - \frac{1}{2}} \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= \frac{1}{4} [arctan (u)]_{- \frac{1}{2}}^{π - \frac{1}{2}}

Compute the boundaries:

arctan (π - \frac{1}{2}) + arctan (\frac{1}{2})

= \frac{1}{4} (arctan (π - \frac{1}{2}) + arctan (\frac{1}{2}))

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