Evaluate the integral: I = integral dx/(1 + cos(x))

Evaluate the integral: $$ I = \int \frac{dx}{1 + \cos(x)}. $$

Step 1 · Apply the half-angle identity

To solve the given integral, we begin by simplifying the integrand. The denominator contains $1+\cos(x)$, which suggests we can use a trigonometric identity to simplify it. We use the following identity:

$$ 1+\cos(x) = 2\cos^2\left(\frac{x}{2}\right). $$

Thus, the integral becomes:

$$ I = \int \frac{dx}{2\cos^2\left(\frac{x}{2}\right)}. $$

We can factor out the constant $\dfrac12$ from the integral:

$$ I = \frac{1}{2}\int \frac{dx}{\cos^2\left(\frac{x}{2}\right)}. $$

Now, we recognize that $\dfrac{1}{\cos^2\left(\frac{x}{2}\right)}=\sec^2\left(\dfrac{x}{2}\right)$, so the integral becomes:

$$ I = \frac{1}{2}\int \sec^2\left(\frac{x}{2}\right)\,dx. $$

Step 2 · Substitute

To simplify further, we make a substitution. Let:

$$ u = \frac{x}{2}. $$

Then:

$$ du = \frac{dx}{2}, \quad dx = 2\,du. $$

Step 3 · Transform and integrate

Substituting into the integral:

$$ I = \frac{1}{2}\int \sec^2(u)\cdot 2\,du = \int \sec^2(u)\,du. $$

The integral of $\sec^2(u)$ is a standard result:

$$ \int \sec^2(u)\,du = \tan(u). $$

Thus, the integral becomes:

$$ I = \tan(u) + C, $$

where $C$ is the constant of integration.

Step 4 · Substitute back

Finally, substituting back $u = \dfrac{x}{2}$, we obtain the solution:

$$ I = \tan\left(\frac{x}{2}\right) + C. $$

Summary Table

Additional Notes

• The domain of $I$ excludes $x=\pi+2k\pi$ for integer $k$, where $1+\cos x = 0$.
• The two constant factors — $\tfrac12$ from the half-angle identity and $2$ from $dx=2\,du$ — exactly cancel, leaving a clean $\int \sec^2u\,du$ with no leftover coefficient.
• The result can be checked by differentiating $\tan(x/2)$ with the chain rule and confirming it returns $\dfrac{1}{1+\cos x}$.