Question

`int_0^1 x^2e^x dx` = ______.

Options

• e – 2

• e + 2

• e² – 2

• e²

Answer 1

`int_0^1 x^2e^x dx` = e – 2.

Explanation:

Let I = `int_0^1 x^2e^x dx`

= `x^2int_0^1 e^x dx - int_0^1[d/(dx) (x^2)inte^x]dx`

`\implies` I = `[x^2.e^x]_0^1 - int_0^1 2x.e^x dx`

= `1.e^1 - 2int_0^1 x  e^x dx`

= `e - 2[x int_0^1 e^x dx - int_0^1 e^x dx]`

= `e - 2[xe^x - e^x]_0^1`

= e – 2[e – e – (0 – 1)]

= e – 2