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

Concept: U Substitution — the denominator $1 + \sin 2x$ simplifies to $(\sin x + \cos x)^2$, which suggests a direct substitution.

Step 1: Use the identity $\sin 2x = 2 \sin x \cos x$. Then

Step 2: Let $u = \sin x + \cos x$. Then

which is exactly the numerator.

Step 3: The integral becomes …

Let’s start with the intuition. When you see an integrand like $\frac{\cos x - \sin x}{1 + \sin 2x}$, your first thought might be to try a standard substitution or a trigonometric identity. The denominator has $\sin 2x$, which is $2 \sin x \cos x$. That suggests we might be able to rewrite $1 + \sin 2x$ as a perfect square. Indeed, recall that $(\cos x + \sin x)^2 = \cos^2 x + \sin^2 x + 2 \sin x \cos x = 1 + \sin 2x$. That’s a clean match.

Now look at the numerator: $\cos x - \sin x$. Notice that the derivative of $\cos x + \sin x$ is $-\sin x + \cos x = \cos x - \sin x$. That’s exactly the numerator! So we have a function and its derivative sitting in the integrand — a classic setup for substitution.

Let’s work through it step by step.

1. Rewrite the denominator Use the identity:

So the integral becomes:

2. Choose a substitution Let $t = \cos x + \sin x$. Then differentiate:

So $dt = (\cos x - \sin x) \, dx$. This matches the numerator exactly.

3. Rewrite the integral in terms of $t$ Substituting, we get:

4. Integrate The integral $\int \frac{dt}{t^2}$ is a standard power rule: …

Method: Recognise $1+\sin 2x=(\sin x+\cos x)^2$ and substitute

When $1+\sin 2x$ appears in a denominator with $\cos x-\sin x$ (or $\cos x+\sin x$) on top, rewrite the denominator as a perfect square so a direct substitution applies.

Steps

Step 1: Rewrite the denominator as a perfect square.

Step 2: Substitute $u=\sin x+\cos x$.

The numerator $\cos x-\sin x$ is exactly $du$ — a clean match. …

Common Mistakes

Mistake 1: Not recognising $1+\sin 2x$ as a perfect square.

Why it's wrong: without seeing $1+\sin 2x=(\sin x+\cos x)^2$, the integrand looks intractable. Correct approach: rewrite the denominator as $(\sin x+\cos x)^2$ so $u=\sin x+\cos x$ works directly.

Mistake 2: Getting the sign of $du$ wrong. …