Question

`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`

Options

• `x"e"^(sin^-1 x) + "c"`

• `x^2"e"^(sin^-1 x) + "c"`

• `1/x "e"^(sin^-1 x) + "c"`

• `1/x^2 "e"^(sin^-1 x) + "c"`

Answer 1

`x"e"^(sin^-1 x) + "c"`

Explanation:

Put `sin^-1 x = "t" => "dx"/sqrt(1 - x^2) = "dt"`

`therefore int "e"^(sin^-1x) ((x + sqrt(1 - x^2))/(sqrt(1 - x^2)))`dx

`= int "e"^"t"[sin "t" + sqrt(1 - sin^2 "t")] "dt"`

`int "e"^"t"(sin "t" + cos "t")`dt

`= "e"^"t" sin "t" + "c"    ....[because int "e"^x ["f" (x) + "f'"(x)]"dx" = "e"^x "f"(x) + "c"]`

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