What is the integral from 3 to 5 of 6x^2(x^3+4) ?

The integral from 3 to 5 of 6x^2(x^3+4) is 15680

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integral from 3 to 5 of 6x^2(x^3+4)

\int_{3}^{5} 6 x^{2} (x^{3} + 4) d x

15680

Solution steps

\int_{3}^{5} 6 x^{2} (x^{3} + 4) d x

Take the constant out:

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

= 6 \cdot \int_{3}^{5} x^{2} (x^{3} + 4) d x

Expand

x^{2} (x^{3} + 4) : x^{5} + 4 x^{2}

= 6 \cdot \int_{3}^{5} x^{5} + 4 x^{2} 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

= 6 (\int_{3}^{5} x^{5} d x + \int_{3}^{5} 4 x^{2} d x)

\int_{3}^{5} x^{5} d x = \frac{7448}{3}

\int_{3}^{5} 4 x^{2} d x = \frac{392}{3}

= 6 (\frac{7448}{3} + \frac{392}{3})

Simplify

6 (\frac{7448}{3} + \frac{392}{3}) : 15680

= 15680

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What is the integral from 3 to 5 of 6x^2(x^3+4) ?
The integral from 3 to 5 of 6x^2(x^3+4) is 15680