What is the integral from 0 to 3 of x/(sqrt(36-x^2)) ?

The integral from 0 to 3 of x/(sqrt(36-x^2)) is 3(2-sqrt(3))

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integral from 0 to 3 of x/(sqrt(36-x^2))

\int_{0}^{3} \frac{x}{36 - x ^{2}} d x

3 (2 - 3)

Decimal

0.80384 \dots

Solution steps

\int_{0}^{3} \frac{x}{36 - x ^{2}} d x

Apply u-substitution

= \int_{36}^{27} - \frac{1}{2 u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{27}^{36} - \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_{27}^{36} \frac{1}{u} d u)

Apply the Power Rule

= - (- \frac{1}{2} [2 u^{\frac{1}{2}}]_{27}^{36})

Simplify

- (- \frac{1}{2} [2 u^{\frac{1}{2}}]_{27}^{36}) : \frac{1}{2} [2 u]_{27}^{36}

= \frac{1}{2} [2 u]_{27}^{36}

Compute the boundaries:

12 - 63

= \frac{1}{2} (12 - 63)

Simplify

= 3 (2 - 3)

What is the integral from 0 to 3 of x/(sqrt(36-x^2)) ?
The integral from 0 to 3 of x/(sqrt(36-x^2)) is 3(2-sqrt(3))