Q.Integrate the following function: $\frac{\sin^3 x + \cos^3 x}{\sin^2 x \cos^2 x}$

Concept: Sine Double Angle Integration — rewriting the integrand using $\sin 2x = 2\sin x \cos x$ to simplify.

Step 1: Split the fraction:

Step 2: Write each term as a secant/tangent or cosecant/cotangent derivative:

$\frac{\sin x}{\cos^2 x} = \sec x \tan x, \quad \frac{\cos x}{\sin^2 x} = \csc x \cot x$ …

1. Split the integrand.

2. Rewrite each term.

3. Integrate using standard results.

Hence …

Method: Split a sum-over-product fraction into standard trig integrals

When a numerator is a sum of terms over a product denominator, divide each numerator term separately — often each piece simplifies to a recognisable $\sec x\tan x$ or $\csc x\cot x$ form.

Steps

Step 1: Split the fraction term by term.

Step 2: Cancel common powers in each piece.

$\frac{\sin x}{\cos^2 x}+\frac{\cos x}{\sin^2 x}=\sec x\tan x+\csc x\cot x$ …

Common Mistakes

Mistake 1: Trying to integrate the fraction whole instead of splitting the sum.

Why it's wrong: the combined form has no obvious substitution, but each half simplifies to a known derivative. Correct approach: split into $\dfrac{\sin^3 x}{\sin^2 x\cos^2 x}+\dfrac{\cos^3 x}{\sin^2 x\cos^2 x}=\sec x\tan x+\csc x\cot x$.

Mistake 2: Sign error on the $\csc x\cot x$ term. …