Question

`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.

Options

• e^x cot + c

• - e^x cot + c

• e^x tan + c

• - e^x tan 2x + c

Answer 1

`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = - e^x cot + c.

Explanation:

`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]  "dx" = int "e"^x [(2 (1 - sin x cos x))/(2 sin^2 x)]`dx

`= e^x (cosec^2 x - cot x)`dx

`= - int "e"^x (- cosec^2 x + cot x)`dx

`= - e^x cot x + "c"   ....[because int "e"^x ["f"(x) + "f'"(x)]"dx" = "e"^x "f"(x) + "c"]`