How ?

integral sinx / 4 + sinx dx

The standard approach is to employ the substitution . Then . You can derive the following relations:



(where the choice of comes from the fact that it satisfies the Pythagorean theorem, )



and we also have



So the integral can be written as



Simplifying the integrand gives



which can be expanded into partial fractions, yielding



The first integral can be handled immediately (), while the other needs rewriting in order to be computed similarly. Complete the square in the denominator to get



and cancelling a factor of 2, you'll end up with



Proper trigonometric substitutions will complete the work. (And of course you'll need to back-substitute to get the result in terms of .)

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What are the choices