what is the answer to ∫tanxsec^2x dx?

Question

i am trying to use U-substitution.
but both of the following work:
u=tanx du=sec^2x
u=secx du=secxtanx

so i would have 2 different answers after pluging U back in....
which is correct?

thanks for answering

Answer 1

∫tanxsec^2xdx will give you two answers. One is tan^2x/2 +c and sec^2 x/2 +c. Both are right coz integration of something is not unique. ^^

Answer 2

you get first tan^2 x/2 +c_1 and second sec^2 x/2+c_2

Now tan^2 x = sec^2 x+1, if you replace this in first result you get
sec^2 x/2 +1/2+c_1, now if you take c_2=1/2+c_1, the enigma is elucidated
So both are correct, because they are the same. You can't have different answers to the same integral.

Answer 3

∫tanxsec^2x dx
let u= tan x du = sec ^2 x dx
∫tanxsec^2x dx=tan^3 x /3 + C

∫tanxsec^2x dx
let u=sec x du =secx tanx dx
∫tanxsec^2x dx=sec^2 x/ 2+C

Answer 4

I = ∫ tan x sec ² x dx
Let u = tan x
du = sec ² x dx
I = ∫ u du
I = u ² / 2 + C
I = ( tan ² x ) / 2 + C