Question

If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.

Options

• x² – 1

• x³ – 1

• 2x² – 1

• 2x³ – 1

Answer 1

If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = x² – 1.

Explanation:

Let I = `int x^3"e"^(x^2)  "d"x`

Put x² = t

⇒ x dx = `"dt"/2`

∴ I = `1/2 int "te"^"t"`

= `1/2 ("te"^"t" - "e"^"t") + "c"`

= `"e"^(x^2)/2 (x^2 - 1) + "c"`

∴ f(x) = x² – 1