What is the integral from-6 to 6 of (t^3-36t)(-20t) ?

The integral from-6 to 6 of (t^3-36t)(-20t) is 41472

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integral from-6 to 6 of (t^3-36t)(-20t)

\int_{- 6}^{6} (t^{3} - 36 t) (- 20 t) d t

41472

Solution steps

\int_{- 6}^{6} (t^{3} - 36 t) (- 20 t) d t

Simplify

= \int_{- 6}^{6} - 20 t (t^{3} - 36 t) d t

Take the constant out:

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

= - 20 \cdot \int_{- 6}^{6} t (t^{3} - 36 t) d t

Expand

t (t^{3} - 36 t) : t^{4} - 36 t^{2}

= - 20 \cdot \int_{- 6}^{6} t^{4} - 36 t^{2} 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

= - 20 (\int_{- 6}^{6} t^{4} d t - \int_{- 6}^{6} 36 t^{2} d t)

\int_{- 6}^{6} t^{4} d t = \frac{15552}{5}

\int_{- 6}^{6} 36 t^{2} d t = 5184

= - 20 (\frac{15552}{5} - 5184)

Simplify

- 20 (\frac{15552}{5} - 5184) : 41472

= 41472

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What is the integral from-6 to 6 of (t^3-36t)(-20t) ?
The integral from-6 to 6 of (t^3-36t)(-20t) is 41472