Question

If $\int \frac{\sin^{3}x + \cos^{3}x}{\sin^{2}x \cos^{2}x} dx = A\sec x + B\csc x + c$ then $(A, B)$ are}

• A. $(1, -1)$
• B. $(-1, -1)$
• C. $(1, 1)$
• D. $(-1, 1)$

Hint:

Always look to split complex fractions with a single term in the denominator.

Answer 1

The Correct Option is A

Step 1: Concept
Split the numerator into two separate fractions and simplify each.

Step 2: Meaning
We obtain integrals of simple trigonometric products.

Step 3: Analysis
$\int \frac{\sin^3 x}{\sin^2 x \cos^2 x} dx + \int \frac{\cos^3 x}{\sin^2 x \cos^2 x} dx = \int \frac{\sin x}{\cos^2 x} dx + \int \frac{\cos x}{\sin^2 x} dx$. These are $\int \tan x \sec x dx + \int \cot x \csc x dx = \sec x - \csc x + c$.

Step 4: Conclusion
Comparing $\sec x - \csc x$ with $A\sec x + B\csc x$, we find $A=1$ and $B=-1$.
Final Answer: (A)