More Substitution Rule — Question 5

Question 5

Evaluate the integral $$\int \frac{dx}{1+\sin x}.$$

Question 5 - Solution

The integrand is defined wherever $1+\sin x\ne0$.

A primitive valid on each such interval is

$$\boxed{-\frac{\cos x}{1+\sin x}+C.}$$

Indeed, the quotient rule gives

$$\frac{d}{dx}\left(-\frac{\cos x}{1+\sin x}\right) =\frac{\sin x(1+\sin x)+\cos^2x}{(1+\sin x)^2} =\frac1{1+\sin x}.$$

This expression introduces no extra excluded points.