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

The integral from 0 to 2 of x/(sqrt(x^2+5)) is 3-sqrt(5)

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integral from 0 to 2 of x/(sqrt(x^2+5))

\int_{0}^{2} \frac{x}{x ^{2} + 5} d x

3 - 5

Decimal

0.76393 \dots

Solution steps

\int_{0}^{2} \frac{x}{x ^{2} + 5} d x

Apply u-substitution

: \int_{5}^{9} \frac{1}{2 u} d u

= \int_{5}^{9} \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_{5}^{9} \frac{1}{u} d u

Apply the Power Rule

: [2 u^{\frac{1}{2}}]_{5}^{9}

= \frac{1}{2} [2 u^{\frac{1}{2}}]_{5}^{9}

Apply radical rule:

a^{\frac{1}{n}} = n a

= \frac{1}{2} [2 u]_{5}^{9}

Compute the boundaries:

6 - 25

= \frac{1}{2} (6 - 25)

Simplify

= 3 - 5

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What is the integral from 0 to 2 of x/(sqrt(x^2+5)) ?
The integral from 0 to 2 of x/(sqrt(x^2+5)) is 3-sqrt(5)