Question

`int ("e"^x(1 + x))/(cos^2 ("e"^x x))"d"x` is equals to ______.

विकल्प

• – cot (ex^x) + C

• tan (xe^x) + C

• tan (e^x) + C

• cot (e^x) + C

Answer 1

`int ("e"^x(1 + x))/(cos^2 ("e"^x x))"d"x` is equals to tan (xe^x) + C.

Explanation:

`int ("e"^x(1 + x))/(cos^2 ("e"^x x))"d"x`

Let xe^x = t

`\implies` (xe^x + e^x) = `("dt")/("d"x)`

`\implies` dx = `("dt")/("e"^x(x + 1))` 

∴ `int ("e"^x(1 + x))/(cos^2("e"^x x))"d"x = int ("e"^x(1 + x))/(cos^2"t") xx ("dt")/("e"^x(1 + x))`

= `int 1/(cos^2"t")"dt"`

= `int sec^2"t"  "dt"`

= tan t + C

= tan (xe^x) + C