What is the integral of #tan(x)#?

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Answer 1 · 2016-02-20T14:35:48

#int tanx "d"x = -ln|cosx| + "constant"#

From the chain rule, we know that

#frac{"d"}{"d"x} (ln(f(x))) = frac{1}{f(x)}*f'(x)#

Therefore,

#int frac{f'(x)}{f(x)} "d"x = ln|f(x)| + "Constant"#

We also know that

#frac{"d"}{"d"x}(cosx) = -sinx#

And that

#tanx = frac{sinx}{cosx}#

# = -frac{-sinx}{cosx}#

#= -frac{frac{"d"}{"d"x}(cosx)}{cosx}#

Hence,

#int tanx "d"x = -ln|cosx| + "constant"#