What is the integral from 0 to 2 of (2x-6)(8x^2+5) ?

The integral from 0 to 2 of (2x-6)(8x^2+5) is -104

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integral from 0 to 2 of (2x-6)(8x^2+5)

\int_{0}^{2} (2 x - 6) (8 x^{2} + 5) d x

- 104

Solution steps

\int_{0}^{2} (2 x - 6) (8 x^{2} + 5) d x

Expand

(2 x - 6) (8 x^{2} + 5) : 16 x^{3} + 10 x - 48 x^{2} - 30

= \int_{0}^{2} 16 x^{3} + 10 x - 48 x^{2} - 30 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}^{2} 16 x^{3} d x + \int_{0}^{2} 10 x d x - \int_{0}^{2} 48 x^{2} d x - \int_{0}^{2} 30 d x

\int_{0}^{2} 16 x^{3} d x = 64

\int_{0}^{2} 10 x d x = 20

\int_{0}^{2} 48 x^{2} d x = 128

\int_{0}^{2} 30 d x = 60

= 64 + 20 - 128 - 60

Simplify

= - 104

\int_{\frac{1}{2}}^{1} (x^{- 3} - 19) d x

\int_{2}^{4} \frac{x}{x - 1} d x

\int_{0}^{4} (x^{2} + 2) d x

\int_{1}^{5} \frac{1}{x ^{2} + 1} d x

\int_{0}^{1} 4 + 5 x d x

What is the integral from 0 to 2 of (2x-6)(8x^2+5) ?
The integral from 0 to 2 of (2x-6)(8x^2+5) is -104