Question

If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is ______.

Options

• e

• 1

• 0

• `1/e`

Answer 1

If `β + 2int_0^1x^2e^(-x^2)dx = int_0^1e^(-x^2)dx`, then the value of β is `underlinebb(1/e)`.

Explanation:

`β - int_0^1xe^(-x^2)(-2x)dx` = `int_0^1e^(-x^2)dx`

⇒ `β - int_0^1\underset(I)(x)\underset(II)((e^(-x^2)))dx` = `int_0^1 e^(-x^2)dx`

⇒ `β - [(xe^(-x^2))_0^1 - int_0^1 e^(-x^2)dx]` = `int_0^1 e^(-x^2)dx`

⇒ `β - 1/e + int_0^1e^(-x^2)dx` = `int_0^1e^(-x^2)dx`

⇒ β = `1/e`