Q.Integrate the following function: $\frac{\sin x}{(1 + \cos x)^2}$

The key idea is U Substitution — we choose a substitution that simplifies the denominator.

Let $u = 1 + \cos x$. Then $du = -\sin x \, dx$, so $\sin x \, dx = -du$.

The integral becomes:

Integrate: …

Why U-Substitution Works Here

When you see a function like $\frac{\sin x}{(1 + \cos x)^2}$, your first instinct should be to look for a function and its derivative hiding inside. The denominator has $(1 + \cos x)^2$, and the numerator has $\sin x$. Notice that the derivative of $\cos x$ is $-\sin x$, and the derivative of $1 + \cos x$ is $-\sin x$. That’s almost a perfect match — we just need to account for the sign.

This is the classic setup for u-substitution: we let $u$ be the "inside" function whose derivative appears (up to a constant factor) in the numerator. Here, $u = 1 + \cos x$ is the natural choice because its derivative $-\sin x$ is right there, just missing a negative sign.

Step-by-Step Solution

1. Choose the substitution.

   Let $u = 1 + \cos x$.

   Then $\frac{du}{dx} = -\sin x$, so $du = -\sin x \, dx$, which means $\sin x \, dx = -du$.

2. Rewrite the integral in terms of $u$.

   The original integral is:

Substituting $u = 1 + \cos x$ and $\sin x \, dx = -du$, we get:

3. Integrate using the power rule. The power rule for integration says $\int u^n \, du = \frac{u^{n+1}}{n+1} + C$ for $n \neq -1$. Here $n = -2$, so:

That simplifies to $\frac{1}{u} + C$. …

Method: Same substitution, but a power rather than a log

Use this when the denominator is raised to a power, e.g. $\dfrac{\sin x}{(1+\cos x)^2}$: the same substitution as the log case, but now you power-rule instead of taking a logarithm.

Steps

Step 1: Substitute the base of the power.

Let $u=1+\cos x$, so $du=-\sin x\,dx$, i.e. $\sin x\,dx=-du$. The integral becomes $-\displaystyle\int u^{-2}\,du$.

Step 2: Apply the power rule (exponent $-2\neq-1$). …

Common Mistakes

Mistake 1: Using a logarithm because the numerator is the denominator's derivative.

Why it's wrong: the denominator is squared, $(1+\cos x)^2$, so the exponent is $-2$, giving a power-rule result, not a log. Correct approach: substitute $u=1+\cos x$ and integrate $u^{-2}$.

Mistake 2: Sign error from $du=-\sin x\,dx$. …