What is the integral from 0 to 3 of x/(sqrt(16+x^2)) ?

The integral from 0 to 3 of x/(sqrt(16+x^2)) is 1

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integral from 0 to 3 of x/(sqrt(16+x^2))

\int_{0}^{3} \frac{x}{16 + x ^{2}} d x

1

Solution steps

\int_{0}^{3} \frac{x}{16 + x ^{2}} d x

Apply u-substitution

: \int_{16}^{25} \frac{1}{2 u} d u

= \int_{16}^{25} \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_{16}^{25} \frac{1}{u} d u

Apply the Power Rule

: [2 u^{\frac{1}{2}}]_{16}^{25}

= \frac{1}{2} [2 u^{\frac{1}{2}}]_{16}^{25}

Apply radical rule:

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

= \frac{1}{2} [2 u]_{16}^{25}

Compute the boundaries:

2

= \frac{1}{2} \cdot 2

Simplify

= 1

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What is the integral from 0 to 3 of x/(sqrt(16+x^2)) ?
The integral from 0 to 3 of x/(sqrt(16+x^2)) is 1