integration of sec ^2 x . tan x = tan^2 x/2??

yep, it's right. I hope that the R.H.S in ur post  is (tan^2x)/2

What's the doubt in such simple question.I can assure you that the answer in the book is wrong.

You can simply observe that d(tanx)=sec²xdx and so,the answer is (tan²x)/2.I don't have any doubt in this answer.

is the answer.but this is also having other answer please rectifiy me if i am wrong.

Ya..Now I understand.I forgot to write the constant of integration in the answer.It can have both the answers and in fact any no. of answers bcoz an indefinite integral has any no. of answers depending on the constant c.

Here,The answer is tan²x/2+c=tan²x+1/2+c-1/2=sec²x/2+c' where c'=c-1/2.

So,both the answers are correct.Sorry for not telling this earlier.It didn't strike to my mind that time.

jauna there can be more than one answer for a quwstion.so please try to get all the answers.

jerrycliff jauna is correct.