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

1 Answer
Jun 7, 2018

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

Explanation:

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