What is the integral from 0 to 5 of 15xsqrt(x+4) ?

The integral from 0 to 5 of 15xsqrt(x+4) is 506

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integral from 0 to 5 of 15xsqrt(x+4)

\int_{0}^{5} 15 x x + 4 d x

506

Solution steps

\int_{0}^{5} 15 x x + 4 d x

Take the constant out:

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

= 15 \cdot \int_{0}^{5} x x + 4 d x

Apply u-substitution

: \int_{4}^{9} (u - 4) u d u

= 15 \cdot \int_{4}^{9} (u - 4) u d u

Expand

(u - 4) u : u^{\frac{3}{2}} - 4 u

= 15 \cdot \int_{4}^{9} u^{\frac{3}{2}} - 4 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

= 15 (\int_{4}^{9} u^{\frac{3}{2}} d u - \int_{4}^{9} 4 u d u)

\int_{4}^{9} u^{\frac{3}{2}} d u = \frac{422}{5}

\int_{4}^{9} 4 u d u = \frac{152}{3}

= 15 (\frac{422}{5} - \frac{152}{3})

Simplify

15 (\frac{422}{5} - \frac{152}{3}) : 506

= 506

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What is the integral from 0 to 5 of 15xsqrt(x+4) ?
The integral from 0 to 5 of 15xsqrt(x+4) is 506