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

The integral from 0 to 2 of (2x)/(x^2+4) is ln(2)

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

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

ln (2)

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int_{4}^{8} \frac{1}{2 u} d u

= 2 \cdot \int_{4}^{8} \frac{1}{2 u} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int_{4}^{8} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} [ln ∣ u ∣]_{4}^{8}

Simplify

2 \cdot \frac{1}{2} [ln ∣ u ∣]_{4}^{8} : [ln ∣ u ∣]_{4}^{8}

= [ln ∣ u ∣]_{4}^{8}

Compute the boundaries:

ln (2)

= ln (2)

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