What is the integral from-1 to 1 of x^2sqrt(5-3x) ?

The integral from-1 to 1 of x^2sqrt(5-3x) is (580sqrt(2))/(567)

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integral from-1 to 1 of x^2sqrt(5-3x)

\int_{- 1}^{1} x^{2} 5 - 3 x d x

\frac{580 2}{567}

Decimal Notation

1.44663 \dots

Solution steps

\int_{- 1}^{1} x^{2} 5 - 3 x d x

Apply u-substitution

: \int_{8}^{2} - \frac{u ( u - 5 ) ^{2}}{27} d u

= \int_{8}^{2} - \frac{u ( u - 5 ) ^{2}}{27} d u

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

= - \int_{2}^{8} - \frac{u ( u - 5 ) ^{2}}{27} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - (- \frac{1}{27} \cdot \int_{2}^{8} u (u - 5)^{2} d u)

Expand

u (u - 5)^{2} : u^{\frac{5}{2}} - 10 u^{\frac{3}{2}} + 25 u

= - (- \frac{1}{27} \cdot \int_{2}^{8} u^{\frac{5}{2}} - 10 u^{\frac{3}{2}} + 25 u d u)

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- \frac{1}{27} (\int_{2}^{8} u^{\frac{5}{2}} d u - \int_{2}^{8} 10 u^{\frac{3}{2}} d u + \int_{2}^{8} 25 u d u))

\int_{2}^{8} u^{\frac{5}{2}} d u = \frac{2032 2}{7}

\int_{2}^{8} 10 u^{\frac{3}{2}} d u = 4962

\int_{2}^{8} 25 u d u = \frac{700 2}{3}

= - (- \frac{1}{27} (\frac{2032 2}{7} - 4962 + \frac{700 2}{3}))

Simplify

- (- \frac{1}{27} (\frac{2032 2}{7} - 4962 + \frac{700 2}{3})) : \frac{580 2}{567}

= \frac{580 2}{567}

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What is the integral from-1 to 1 of x^2sqrt(5-3x) ?
The integral from-1 to 1 of x^2sqrt(5-3x) is (580sqrt(2))/(567)