What is the integral from 0 to 3 of 10t(75-t^2) ?

The integral from 0 to 3 of 10t(75-t^2) is 6345/2

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integral from 0 to 3 of 10t(75-t^2)

\int_{0}^{3} 10 t (75 - t^{2}) d t

\frac{6345}{2}

Decimal

3172.5

Solution steps

\int_{0}^{3} 10 t (75 - t^{2}) d t

Take the constant out:

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

= 10 \cdot \int_{0}^{3} t (75 - t^{2}) d t

Expand

t (75 - t^{2}) : 75 t - t^{3}

= 10 \cdot \int_{0}^{3} 75 t - t^{3} d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 10 (\int_{0}^{3} 75 t d t - \int_{0}^{3} t^{3} d t)

\int_{0}^{3} 75 t d t = \frac{675}{2}

\int_{0}^{3} t^{3} d t = \frac{81}{4}

= 10 (\frac{675}{2} - \frac{81}{4})

Simplify

10 (\frac{675}{2} - \frac{81}{4}) : \frac{6345}{2}

= \frac{6345}{2}

What is the integral from 0 to 3 of 10t(75-t^2) ?
The integral from 0 to 3 of 10t(75-t^2) is 6345/2