What is the integral from 3 to 5 of x^3sqrt(x^2-4) ?

The integral from 3 to 5 of x^3sqrt(x^2-4) is (441sqrt(21))/5-5sqrt(5)+4(7sqrt(21)-(5sqrt(5))/3)

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integral from 3 to 5 of x^3sqrt(x^2-4)

\int_{3}^{5} x^{3} x^{2} - 4 d x

\frac{441 21}{5} - 55 + 4 (721 - \frac{5 5}{3})

Decimal

506.40783 \dots

Solution steps

\int_{3}^{5} x^{3} x^{2} - 4 d x

Apply u-substitution

: \int_{5}^{21} (u^{2} + 4) u^{2} d u

= \int_{5}^{21} (u^{2} + 4) u^{2} d u

Expand

(u^{2} + 4) u^{2} : u^{4} + 4 u^{2}

= \int_{5}^{21} u^{4} + 4 u^{2} 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

= \int_{5}^{21} u^{4} d u + \int_{5}^{21} 4 u^{2} d u

\int_{5}^{21} u^{4} d u = \frac{441 21}{5} - 55

\int_{5}^{21} 4 u^{2} d u = 4 (721 - \frac{5 5}{3})

= \frac{441 21}{5} - 55 + 4 (721 - \frac{5 5}{3})

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What is the integral from 3 to 5 of x^3sqrt(x^2-4) ?
The integral from 3 to 5 of x^3sqrt(x^2-4) is (441sqrt(21))/5-5sqrt(5)+4(7sqrt(21)-(5sqrt(5))/3)