What is the antiderivative of $(sinx + cosx)/tanx$ ?

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Answer 1 · 2016-08-21T23:19:59

$sinx-ln(abs(cotx+cscx))+cosx+C$

We have:

$int(sinx+cosx)/tanxdx=int(sinx+cosx)/(sinx/cosx)dx=int(cosx(sinx+cosx))/sinxdx$

$=intcosxdx+intcos^2x/sinxdx$

The first is a well known integral. For the second term, rewrite $cos^2x$ using the Pythagorean Identity:

$=sinx+int(1-sin^2x)/sinxdx=sinx+intcscxdx-intsinxdx$

Again, these are both well-known integrals:

$=sinx-ln(abs(cotx+cscx))+cosx+C$

What is the antiderivative of (sinx + cosx)/tanx(sinx + cosx)/tanx?