Q.$\int \dfrac{x+\sin x}{1+\cos x}\,dx$ is equal to
(A) $\log|1+\cos x| + C$
(B) $\log|x+\sin x| + C$
(C) $x - \tan\dfrac{x}{2} + C$
(D) $x\tan\dfrac{x}{2} + C$

Concept: U Substitution — rewrite the integrand using the identity $1+\cos x = 2\cos^2\frac{x}{2}$ and split the fraction.

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

Step 2:

Step 3: Integrate term by term:

(using integration by parts with $u=x$, $dv=\frac12\sec^2\frac{x}{2}dx$). …

We start with the integral

The denominator $1+\cos x$ is a classic trigger for the half-angle identity: $1+\cos x = 2\cos^2\frac{x}{2}$. This identity will let us break the fraction into manageable pieces.

Why this approach works

A direct substitution like $u = 1+\cos x$ fails because the numerator has an $x$ term, not just $\sin x$. Instead, we split the integrand:

The second term is a standard form; the first term, after using the half-angle identity, becomes $x \cdot \frac{1}{2}\sec^2\frac{x}{2}$, which is a perfect candidate for integration by parts (or a clever observation about derivatives).

Step-by-step solution

1. Rewrite the denominator using $1+\cos x = 2\cos^2\frac{x}{2}$. Then

2. Split the integral:

3. First integral — use the half-angle form:

Notice that $\frac{d}{dx}\left(\tan\frac{x}{2}\right) = \frac{1}{2}\sec^2\frac{x}{2}$. So the integrand is $x$ times the derivative of $\tan\frac{x}{2}$. This suggests integration by parts, but there’s a cleaner observation:

Rearranging,

Therefore,

4. Second integral — simplify $\frac{\sin x}{1+\cos x}$. Using $\sin x = 2\sin\frac{x}{2}\cos\frac{x}{2}$ and $1+\cos x = 2\cos^2\frac{x}{2}$, …

Method: Half-angle identity plus a reverse product rule

For $\int\dfrac{x+\sin x}{1+\cos x}\,dx$-type integrands, use $1+\cos x=2\cos^2\tfrac x2$ to split into an $x$-piece and a plain $\tan\tfrac x2$ piece.

Steps

Step 1: Rewrite with half-angles.

$\dfrac{1}{1+\cos x}=\tfrac12\sec^2\tfrac x2$ and $\dfrac{\sin x}{1+\cos x}=\tan\tfrac x2$, so the integrand $=\dfrac x2\sec^2\tfrac x2+\tan\tfrac x2$.

Step 2: Recognise the product-rule derivative. …

Common Mistakes

Mistake 1: Substituting $u=1+\cos x$ on the whole fraction.

Why it's wrong: it works for $\frac{\sin x}{1+\cos x}$ but fails for the $x$ term, since $x$ is not a function of $u$. Correct approach: split the integrand and use the half-angle identity first.

Mistake 2: Missing the reverse product-rule cancellation. …