Question

Why is the integral of tan x dx ln(sec x)+C?

Answer 1

\[\int\limits \tan x = \ln \left| \sec x \right| + C\]
i thought it was ln (cos x) + C ?

Answer 2

well, the integral of \[\Large \int \tan(x)dx=-\ln(\cos(x))+C \]

Answer 3

yeah oh whoops forgot the negative.

Answer 4

but you can also apply laws of logarithms to this expression. And that's how you get the sec(x) expression.

Answer 5

log rules? how?

Answer 6

\[\Large a\log(b)=\log(b^a) \]

Answer 7

sorry, can you explain how it -ln(cosx) became sec x ?

Answer 8

sure:
\[\Large -\ln(\cos x)=-1\ln(\cos x)=\ln(\cos x^{-1})=\ln\left(\frac{1}{\cos x}\right) \]

Answer 9

so what is 1/cos(x)?

Answer 10

ohhh
ok thanks
yeah i got it :)

Answer 11

great, well done
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