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What is the antiderivative of #1/(2+sinx)#?

Answer 1

Aurora Alvarado

#I=2/sqrt3tan^-1((2tan(x/2)+1)/sqrt3)+c#

#I=int1/(2+sinx)dx#

Subst. #tan(x/2)=t=>sec^2(x/2)*1/2dx=dt#

#=>[1+tan^2(x/2)]dx=2dt=>dx=(2dt)/(1+t^2)#

#I=int1/(2+(2t)/(1+t^2))xx(2dt)/(1+t^2)#

#I=int1/(2+2t^2+2t)xx2dt#

#=int1/(t^2+t+1)dt#

#=int1/(t^2+t+1/4 +3/4)dt#

#=int1/((t+1/2)^2+(sqrt3/2)^2)dt#

But, #color(red)(int1/(x^2+a^2)dx=1/atan^-1(x/a)+c#

#:.I=1/(sqrt3/2)tan^-1((t+1/2)/(sqrt3/2))+c#

#I=2/sqrt3tan^-1((2t+1)/sqrt3)+c#

Subst. back , #t=tan(x/2)# ,we get

#I=2/sqrt3tan^-1((2tan(x/2)+1)/sqrt3)+c#

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Answer 2

Emma Aldridge

The antiderivative of ( \frac{1}{2+\sin(x)} ) is ( -\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{\tan\left(\frac{x}{2}\right)+\sqrt{3}}{\sqrt{3}-\tan\left(\frac{x}{2}\right)}\right)+C ), where ( C ) is the constant of integration.

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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.