What is the integral from 0 to 15 of xsqrt(15-x) ?

The integral from 0 to 15 of xsqrt(15-x) is 60sqrt(15)

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integral from 0 to 15 of xsqrt(15-x)

\int_{0}^{15} x 15 - x d x

6015

Decimal

232.37900 \dots

Solution steps

\int_{0}^{15} x 15 - x d x

Apply u-substitution

= \int_{15}^{0} - (- u + 15) u d u

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

= - \int_{0}^{15} - (- u + 15) u d u

Take the constant out:

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

= - (- \int_{0}^{15} (- u + 15) u d u)

Expand

(- u + 15) u : - u^{\frac{3}{2}} + 15 u

= - (- \int_{0}^{15} - u^{\frac{3}{2}} + 15 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

= - (- (- \int_{0}^{15} u^{\frac{3}{2}} d u + \int_{0}^{15} 15 u d u))

\int_{0}^{15} u^{\frac{3}{2}} d u = 9015

\int_{0}^{15} 15 u d u = 15015

= - (- (- 9015 + 15015))

Simplify

= 6015

What is the integral from 0 to 15 of xsqrt(15-x) ?
The integral from 0 to 15 of xsqrt(15-x) is 60sqrt(15)