Question about the integral of tangent?

Question

how can the integral of tan(x) = -ln(sec(x)) ? isnt it suppose to equal to -ln(cos(x))...the derivative of cosine is -sin(x) so I would expect it to be ln(f(x))

Answer 1

∫ tanx dx =

rewrite it in terms of sinx, cosx:

∫ (sinx/cosx) dx =

being (- sinx) the derivative of cosx

∫ - [d(cosx)] /cosx = - ln | cosx | + C

if you want it in terms of secx, you just have to change the sign as:

∫ tanx dx = - ln | cosx | + C = ln | secx | + C =

(recall the rule: ln a = - ln (1/a))

I hope it helps...

Bye!