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

The integral from 0 to 2 of x/((1+x^2)) is 1/2 ln(5)

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

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

\frac{1}{2} ln (5)

Decimal

0.80471 \dots

Solution steps

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

Apply u-substitution

= \int_{1}^{5} \frac{1}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{1}^{5} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} [ln ∣ u ∣]_{1}^{5}

Compute the boundaries:

ln (5)

= \frac{1}{2} ln (5)

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