How can I find the integral of cosx/sinx+cosx?

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Answer 1 · 2018-01-22T20:36:59

#intcosx/sinx+cosxdx=lnabssinx+sinx+C#

Given: #intcosx/sinx+cosxdx#

We are essentially finding two integrals so we really have

#intcosx/sinxdx+intcosdx#

We'll solve each integral separately

#intcosx/sinxdx#

Make a substitution

Let #u=sinx=>du=cosxdx#

#int1/udu#

Use: #color(blue)(int1/udu=lnabsu#

#=lnabsu#

Reverse the substitution:

#=lnabssinx#

For #intcosxdx#

This is a common integral whose antiderivative is #sinx#

Putting it all together

#intcosx/sinx+cosxdx=lnabssinx+sinx+C#