Q.Integrate the following function: $\frac{\cos x}{1 + \cos x}$

Concept: U Substitution — rewrite the integrand using a trigonometric identity to simplify the denominator.

Step 1: Use the identity $\cos x = 2\cos^2\frac{x}{2} - 1$ and $1 + \cos x = 2\cos^2\frac{x}{2}$.

Step 2: The integral becomes

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

Why This Approach Works

When you see a rational function of $\cos x$, the standard trick is to use the half-angle substitution $t = \tan(x/2)$. That works, but it can get messy. Here, there's a cleaner path: rewrite $\cos x$ in terms of $\cos(x/2)$ using the double-angle identity. This turns the denominator into something that cancels nicely, leaving you with a simple sum of two terms — one constant, one a standard trigonometric integral.

The intuition: the denominator $1 + \cos x$ is exactly $2\cos^2(x/2)$. That's the key simplification. Once you see that, the rest is straightforward.

Step-by-Step Solution

1. Rewrite the denominator using the half-angle identity. Recall that $\cos x = 2\cos^2(x/2) - 1$. Therefore:

This is the crucial simplification — the denominator becomes a perfect square of a cosine.

2. Rewrite the numerator in terms of $\cos(x/2)$ as well. The numerator is $\cos x$. Using the same identity:

So the integrand becomes:

3. Split the fraction into two simpler terms. Divide each term in the numerator by the denominator:

Now the integrand is expressed as a constant minus a constant times $\sec^2$ of half the angle.

4. Integrate term by term. The integral becomes: $\int \left(1 - \frac{1}{2}\sec^2(x/2)\right) dx = \int 1\,dx - \frac{1}{2}\int \sec^2(x/2)\,dx.$ …

Method: Split off a constant using a half-angle identity

For a rational function of $\cos x$ like $\dfrac{\cos x}{1+\cos x}$, rewrite so the denominator becomes $2\cos^2(x/2)$, then separate the fraction into a constant plus a $\sec^2$ term.

Steps

Step 1: Simplify the denominator.

Step 2: Express the numerator in the same half-angle terms and split.

Writing $\cos x=2\cos^2(x/2)-1$ gives

Step 3: Integrate each piece. …

Common Mistakes

Mistake 1: Cancelling $\dfrac{\cos x}{1+\cos x}$ to $\dfrac{1}{1+1}$ or similar.

Why it's wrong: you cannot cancel a $\cos x$ term that is added, not multiplied, in the denominator. Correct approach: rewrite $1+\cos x=2\cos^2(x/2)$ and split the fraction into $1-\tfrac12\sec^2(x/2)$.

Mistake 2: Dropping the $\tfrac12$ or the argument factor when integrating $\sec^2(x/2)$. …