Question

Int(tan⁡√x/√x)?

Answer 1

The simplified integral is `ln|sec^2(sqrtx)|+C`.

Explanation:

`color(white)=int` `tan(sqrtx)/sqrtx` `dx`

Let `u=sqrtx`, which means `du=1/(2sqrtx)` `dx` and `dx=2sqrtx` `du`:

`=int` `tan(u)/sqrtx` `2sqrtx` `du`

`=int` `tan(u)/color(red)cancelcolor(black)sqrtx` `2color(red)cancelcolor(black)sqrtx` `du`

`=int` `tan(u)*2` `du`

`=2int` `tan(u)` `du`

`=2int` `sin(u)/cos(u)` `du`

Let `w=cos(u)`, which means `dw=-sin(u)` `du` and `du=-(dw)/sin(u)`:

`=2int` `sin(u)/w*-(dw)/sin(u)`

`=2int` `color(red)cancelcolor(black)sin(u)/w*-(dw)/color(red)cancelcolor(black)sin(u)`

`=-2int` `1/w` `dw`

`=-2ln|w|+C`

`=-2ln|cos(u)|+C`

`=-2ln|cos(sqrtx)|+C`

If you want, you can bring the `-2` into the natural log as an exponent:

`=ln|(cos(sqrtx))^-2|+C`

`=ln|1/(cos(sqrtx))^2|+C`

`=ln|1/(cos^2(sqrtx))|+C`

`=ln|sec^2(sqrtx)|+C`

That's the solved integral. Hope this helped!