What is the integral from 1 to 4 of 5sqrt(x)(x-1) ?

The integral from 1 to 4 of 5sqrt(x)(x-1) is 116/3

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integral from 1 to 4 of 5sqrt(x)(x-1)

\int_{1}^{4} 5 x (x - 1) d x

\frac{116}{3}

Decimal Notation

38.66666 \dots

Solution steps

\int_{1}^{4} 5 x (x - 1) d x

Take the constant out:

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

= 5 \cdot \int_{1}^{4} x (x - 1) d x

Expand

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

= 5 \cdot \int_{1}^{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

= 5 (\int_{1}^{4} x^{\frac{3}{2}} d x - \int_{1}^{4} x d x)

\int_{1}^{4} x^{\frac{3}{2}} d x = \frac{62}{5}

\int_{1}^{4} x d x = \frac{14}{3}

= 5 (\frac{62}{5} - \frac{14}{3})

Simplify

5 (\frac{62}{5} - \frac{14}{3}) : \frac{116}{3}

= \frac{116}{3}

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What is the integral from 1 to 4 of 5sqrt(x)(x-1) ?
The integral from 1 to 4 of 5sqrt(x)(x-1) is 116/3