I have a question concerning the integral of: (Secx)^2*(tanx). Now I get the correct answer using integration by parts: u = Secx; du = (Secx)(tanx)dx. This gives me the correct answer to the integral as (1/2)*(Secx)^2+C. The problem is that I can formulate the problem a second way, and get an incorrect answer, but I don't know why. The second (incorrect) way is by letting u = tanx; du = (secx)^2*dx. Then I get (1/2)(tanx)^2 + C which is wrong. But, what am I doing wrong. NOTE: I'm assuming d(tanx)/dx = (secx)^2, and d(secx)/dx = secx*tanx.

Question

I have a question concerning the integral of:  (Secx)^2*(tanx).  Now I get the correct answer using integration by parts:  u = Secx; du = (Secx)(tanx)dx.  This gives me the correct answer to the integral as (1/2)*(Secx)^2+C.  The problem is that I can formulate the problem a second way, and get an incorrect answer, but I don't know why.  The second (incorrect) way is by letting u = tanx; du = (secx)^2*dx.  Then I get (1/2)(tanx)^2 + C which is wrong.  But, what am I doing wrong.  NOTE:  I'm assuming d(tanx)/dx = (secx)^2, and d(secx)/dx = secx*tanx.

anonymous · Answer

Actually both answers are correct.  Since (sec x)^2 = 1 + (tan x)^2, (1/2)*(sec x)^2 + C = (1/2)*(tan x)^2 + C + 1/2 = (1/2)*(tan x)^2 + D (D being a constant).

anonymous · Answer

Thanks very much for the explanation.  Didn't think of that.  Much appreciated.