What is the integral from 1 to 3 of 1/(4+(x-1)^2) ?

The integral from 1 to 3 of 1/(4+(x-1)^2) is pi/8

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integral from 1 to 3 of 1/(4+(x-1)^2)

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

\frac{π}{8}

Decimal

0.39269 \dots

Solution steps

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

Apply u-substitution

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

Apply Integral Substitution

= \int_{0}^{1} \frac{1}{2 ( v ^{2} + 1 )} d v

Take the constant out:

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

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

Use the common integral:

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

= \frac{1}{2} [arctan (v)]_{0}^{1}

Compute the boundaries:

\frac{π}{4}

= \frac{1}{2} \cdot \frac{π}{4}

Simplify

= \frac{π}{8}

What is the integral from 1 to 3 of 1/(4+(x-1)^2) ?
The integral from 1 to 3 of 1/(4+(x-1)^2) is pi/8