MATCH THE FOLLOWING:
(A) $\int \frac{dx}{\sqrt{x(x + 9)}} =$
(B) $\int e^x(1 - \cot x + \cot^2 x) dx =$
(C) $\int \frac{\sin^3 x + \cos^3 x}{\cos^2 x \sin^2 x} dx =$
(D) $\int \frac{dx}{1 - \cos x - \sin x} =$
Step-by-Step Solution
Key Concept: Recognizing when to apply substitution, product rule reversal, trigonometric expansion, or half-angle formulas based on integrand structure.
This question asks to match integrals with their solutions. Option (A) evaluates $\int \frac{dx}{\sqrt{x(x+9)}}$ using substitution $x = t^2$ to get $\frac{2}{3}\tan^{-1}\left(\sqrt{\frac{x}{3}}\right) + c$. Option (B) simplifies $\int e^x(1-\cos x + \cos^2 x)dx$ using product rule recognition to obtain $-e^x\cos x + c$. Option (C) evaluates $\int \frac{\sin^3 x + \cos^3 x}{\cos^2 x \sin^2 x}dx$ by expanding and integrating to get $\sec x - \csc x + c$. Option (D) uses half-angle substitutions to transform $\int \frac{dx}{1-\cos x - \sin x}$ into a standard form yielding $\ln|\cot(x/2)| + c$.
Correct Answer: [A-r] [B-s] [C-q] [D-p]