19-42 evaluate the integral. 23. \int_(0)^(2)(2 x-3)\left(4 x^(2)+1\right) d x

Question

19-42 evaluate the integral.
23.
`\int_(0)^(2)(2 x-3)\left(4 x^(2)+1\right) d x`

Answer 1

Expand the integrand:

`(2x - 3) (4x^2 + 1) = 8x^3 - 12x^2 + 2x - 3`

Then integrate using the power rule:

`\displaystyle \int (8x^3-12x^2+2x-3) \, dx = 2x^4 - 4x^3 + x^2 - 3x + C`

By the fundamental theorem of calculus, the definite integral has a value of

`\displaystyle \int_0^2 (2x-3)(4x^2+1) \, dx = \left(2x^4 - 4x^3 + x^2 - 3x\right)\bigg|_(x=2) - \left(2x^4 - 4x^3 + x^2 - 3x\right)\bigg|_(x=0)`

`\displaystyle \int_0^2 (2x-3)(4x^2+1) \, dx = 32 - 32 + 4 - 6 = \boxed{-2}`