What is the integral from 0 to 5 of x^3sqrt(x^2+25) ?

The integral from 0 to 5 of x^3sqrt(x^2+25) is 2500sqrt(2)-625-(25(250sqrt(2)-125))/3

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integral from 0 to 5 of x^3sqrt(x^2+25)

\int_{0}^{5} x^{3} x^{2} + 25 d x

25002 - 625 - \frac{25 ( 250 2 - 125 )}{3}

Decimal

1005.92231 \dots

Solution steps

\int_{0}^{5} x^{3} x^{2} + 25 d x

Apply u-substitution

= \int_{5}^{52} (u^{2} - 25) u^{2} d u

Expand

(u^{2} - 25) u^{2} : u^{4} - 25 u^{2}

= \int_{5}^{52} u^{4} - 25 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}^{52} u^{4} d u - \int_{5}^{52} 25 u^{2} d u

\int_{5}^{52} u^{4} d u = 25002 - 625

\int_{5}^{52} 25 u^{2} d u = \frac{25 ( 250 2 - 125 )}{3}

= 25002 - 625 - \frac{25 ( 250 2 - 125 )}{3}

What is the integral from 0 to 5 of x^3sqrt(x^2+25) ?
The integral from 0 to 5 of x^3sqrt(x^2+25) is 2500sqrt(2)-625-(25(250sqrt(2)-125))/3