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

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

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

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

5 - 1

Decimal

1.23606 \dots

Solution steps

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

Apply u-substitution

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

= \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

Apply the Power Rule

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

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

Apply radical rule:

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

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

Compute the boundaries:

25 - 2

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

Simplify

= 5 - 1

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\int_{0}^{0.3} e^{- x^{2}} d x

\int_{5}^{0} ∣ 2 x - 5 ∣ d x

\int_{0}^{y} 1 - y^{2} d y

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