I have this problem for homework:
Evaluate the indefinite integral
I don't know how to decide which numbers to pick as the 'u'. Is there a rule to follow, or do you just pick any one at random?
Thank you
$\displaystyle \displaystyle \frac{1}{8} \int \frac{\cos{x}}{\sin{x}+6} \, dx$
$\displaystyle u = \sin{x} + 6$
$\displaystyle du = \cos{x} \, dx$
substitute, integrate, then back substitute ...
$\displaystyle \displaystyle \frac{1}{8} \int \frac{du}{u}$
$\displaystyle \displaystyle \frac{1}{8} \ln{u} + C$
$\displaystyle \displaystyle \frac{1}{8} \ln(\sin{x}+6) + C$
you should already know the following general antiderivative ...
$\displaystyle \displaystyle \int \frac{dx}{x} = \ln|x| + C$
link ...
Pauls Online Notes : Calculus I - Computing Indefinite Integrals