Question

`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.

Options

• `tan(x"e"^x) + "c"`

• `cot(x"e"^x) + "c"`

• `-tan(x"e"^x) + "c"`

• `-cot(x"e"^x) + "c"`

Answer 1

`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = `-cot(x"e"^x) + "c"`.

Explanation:

`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x = int"e"^x (x + 1)"cosec"^2 (x"e"^x)"d"x`

Put xe^x = t

⇒ (x + 1)e^x dx = dt

∴ `int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x = int "cosec"^2"t"  "dt" = - cot "t" + "c"`

= `- cot (x"e"^x) + "c"`