Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
MATCH THE FOLLOWING:
(A) If $\int \frac{2^x}{\sqrt{1-4^x}} dx = k \sin^{-1}(f(x)) + C$, then k is greater than
(B) If $\int \frac{(\sqrt{x})^5}{(\sqrt{x})^7 + x^6} dx = a \ln \frac{x^k}{x^k + 1} + c$, then ak is less than
(C) $\int \frac{x^3 + 1}{x(x^2 + 1)} dx = k \ln|x| + \frac{m}{1 + x^2} + n$, where n is the constant of integration, then mk is greater than
(D) $\int \frac{dx}{5 + 4\cos x} = k \tan^{-1}\left(m \tan \frac{x}{2}\right) + C$, then km is greater than
Step-by-Step Solution
Key Concept: Substitution transforms a complex radical expression into a simple rational integral of the form $\frac{1}{1+y}$.
For part (B), substitute $y = (\sqrt{x})^{-5}$ with $dy/dx = -\frac{5}{2(\sqrt{x})^7}$ to transform the integral into $\int \frac{2dy}{5(1+y)}$, which evaluates to $\frac{2}{5}\ln|1+y| + C = \frac{2}{5}\ln\left(1 + (\sqrt{x})^{-5}\right) + C$. The key steps involve recognizing the power of $\sqrt{x}$ in the denominator and using the substitution to reduce to a standard logarithmic form.
Correct Answer: [A-p, q] [B-r, s] [C-p]