What is the integral from 1 to 3 of 15xsqrt(x+3) ?

The integral from 1 to 3 of 15xsqrt(x+3) is 36sqrt(6)+48

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integral from 1 to 3 of 15xsqrt(x+3)

\int_{1}^{3} 15 x x + 3 d x

366 + 48

Solution steps

\int_{1}^{3} 15 x x + 3 d x

Take the constant out:

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

= 15 \cdot \int_{1}^{3} x x + 3 d x

Apply u-substitution

: \int_{4}^{6} (u - 3) u d u

= 15 \cdot \int_{4}^{6} (u - 3) u d u

Expand

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

= 15 \cdot \int_{4}^{6} u^{\frac{3}{2}} - 3 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}^{6} u^{\frac{3}{2}} d u - \int_{4}^{6} 3 u d u)

\int_{4}^{6} u^{\frac{3}{2}} d u = \frac{72 6 - 64}{5}

\int_{4}^{6} 3 u d u = 126 - 16

= 15 (\frac{72 6 - 64}{5} - (126 - 16))

Simplify

15 (\frac{72 6 - 64}{5} - (126 - 16)) : 366 + 48

= 366 + 48

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What is the integral from 1 to 3 of 15xsqrt(x+3) ?
The integral from 1 to 3 of 15xsqrt(x+3) is 36sqrt(6)+48