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How do you find the antiderivative of #sinx/(1+cosx)#?

Answer 1

Luke Alvarez

#- ln (1 + cos x)#

Finding the pattern is the simplest method.

we know from the chain rule that #d/dx ln(f(x)) = 1/(f(x) )* f'(x)#

here we have #(-f'(x)) / f(x)# because #(1 + cos x)' = - sin x#

so the answer is simply #- ln (1 + cos x)#

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Answer 2

Ezekiel Alexander

To find the antiderivative of ( \frac{\sin(x)}{1 + \cos(x)} ), you can use a substitution. Let ( u = 1 + \cos(x) ). Then, ( du = -\sin(x) , dx ). Substituting, the integral becomes ( -\int \frac{1}{u} , du ), which integrates to ( -\ln|u| + C ), where ( C ) is the constant of integration. Finally, substitute ( u ) back in to get ( -\ln|1 + \cos(x)| + C ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.