Question

How do you evaluate the definite integral `int (6x^2 - 4e^(2x))dx` from `[0,2]`?

Answer 1

`-2(e^4 - 9)~~-91.1963`

Explanation:

Given: `int_0^2 (6x^2 - 4e^(2x)) dx`

Break the integral into two pieces and solve separately:

`int_0^2 6x^2 dx - int_0^2 4e^(2x) dx =`

`6 int_0^2 x^2 dx -4 int_0^2 e^(2x) dx`

Integration rules:
Use the Power rule: `int x^n dx = 1/(n+1) x ^(n+1) + C`

Use `int e^u du = e^u + C`

Let `u = 2x; " "du = 2 dx; " "dx = (du)/2`
`x= 0, => u = 0; " " x= 2 => u = 4`

Using the rules:
`6 int_0^2 x^2 dx -4 int_0^2 e^(2x) dx =`

`6 * 1/3 x^3|_0^2 - 4 int_0^4 e^u(du)/2 =`

`2x^3|_0^2 - 2 int_0^4 e^u du =`

`2(2)^3 - 0 - 2e^(u)|_0^4 =`

`16 - 2(e^4 - e^0) = 16 - 2(e^4 - 1) =`

`16 - 2e^4 +2 = 18 - 2e^4 =`

`-2(e^4 - 9)~~-91.1963`