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

Concept: Trigonometric Simplification — simplify the numerator using identities before integrating.

First, rewrite $\cos 2x = \cos^2 x - \sin^2 x$. Then the numerator becomes:

So the integrand simplifies to: …

Simplify the numerator. With the identity $\cos 2x = 1 - 2\sin^2 x$,

Rewrite the integrand. Dividing by $\cos^2 x$, …

Method: Collapse the numerator with a double-angle identity before dividing

If a numerator combines $\cos 2x$ with $\sin^2 x$ or $\cos^2 x$, substitute the double-angle form so the numerator simplifies (often to a constant), leaving a trivial integral.

Steps

Step 1: Replace $\cos 2x$ with the form that cancels the other term.

Since the numerator has $+2\sin^2 x$, use

so $\cos 2x+2\sin^2 x=1$.

Step 2: Simplify the whole fraction.

$\frac{\cos 2x+2\sin^2 x}{\cos^2 x}=\frac{1}{\cos^2 x}=\sec^2 x$ …

Common Mistakes

Mistake 1: Choosing the wrong form of the $\cos 2x$ identity.

Why it's wrong: to cancel the $+2\sin^2 x$ you need $\cos 2x=1-2\sin^2 x$; using $2\cos^2 x-1$ leaves an uncancelled $\cos^2 x$ term and misses the clean simplification. Correct approach: pick the identity that makes $\cos 2x+2\sin^2 x=1$.

Mistake 2: Overcomplicating a fraction that reduces to $\sec^2 x$. …