What is the integral from 0 to 4 of (sqrt(x)(x-1)) ?

The integral from 0 to 4 of (sqrt(x)(x-1)) is 112/15

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integral from 0 to 4 of (sqrt(x)(x-1))

\int_{0}^{4} (x (x - 1)) d x

\frac{112}{15}

Decimal

7.46666 \dots

Solution steps

\int_{0}^{4} x (x - 1) d x

Expand

x (x - 1) : x^{\frac{3}{2}} - x

= \int_{0}^{4} x^{\frac{3}{2}} - x d x

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}^{4} x^{\frac{3}{2}} d x - \int_{0}^{4} x d x

\int_{0}^{4} x^{\frac{3}{2}} d x = \frac{64}{5}

\int_{0}^{4} x d x = \frac{16}{3}

= \frac{64}{5} - \frac{16}{3}

Simplify

\frac{64}{5} - \frac{16}{3} : \frac{112}{15}

= \frac{112}{15}

What is the integral from 0 to 4 of (sqrt(x)(x-1)) ?
The integral from 0 to 4 of (sqrt(x)(x-1)) is 112/15