What is the integral from 0 to 3 of xsqrt(11-x^2) ?

The integral from 0 to 3 of xsqrt(11-x^2) is (11sqrt(11)-2sqrt(2))/3

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integral from 0 to 3 of xsqrt(11-x^2)

\int_{0}^{3} x 11 - x^{2} d x

\frac{11 11 - 2 2}{3}

Decimal Notation

11.21814 \dots

Solution steps

\int_{0}^{3} x 11 - x^{2} d x

Apply u-substitution

: \int_{11}^{2} - \frac{u}{2} d u

= \int_{11}^{2} - \frac{u}{2} d u

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

= - \int_{2}^{11} - \frac{u}{2} 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_{2}^{11} u d u)

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{2}^{11}

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{2}^{11})

Simplify

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{2}^{11}

Compute the boundaries:

\frac{22 11 - 4 2}{3}

= \frac{1}{2} \cdot \frac{22 11 - 4 2}{3}

Simplify

= \frac{11 11 - 2 2}{3}

\frac{d}{d x} ((\frac{2}{5})^{x})

\int \frac{1}{1 + 4 x ^{2}} d x

(x + y + 4) d x + (2 x + 2 y + 1) d y = 0

\frac{d y}{d x} (25 - x^{2}) + 5 x y = 0

\frac{d}{d x} (\frac{( 1 - x ) ^{2}}{x ^{2}})

What is the integral from 0 to 3 of xsqrt(11-x^2) ?
The integral from 0 to 3 of xsqrt(11-x^2) is (11sqrt(11)-2sqrt(2))/3