Int(tan⁡√x/√x)?

1 Answer
Apr 15, 2018

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!