Integral Calculus-1
Integral Calculus-1
Allen Star Batch
Grade 12
Question:
If $f(x) = \lim_{n \to \infty} \frac{x^n - x^{-n}}{x^n + x^{-n}}$, $0 < x < 1$, $n \in \mathbb{N}$ then $\int (\sin^{-1} x) f'(x) dx$ is equal to:
$-\left[x\sin^{-1} x + \sqrt{1-x^2}\right] + C$
$x\sin^{-1} x + \sqrt{1-x^2} + C$
$\frac{x^2}{2} + C$
$\frac{1}{2}(\sin^{-1} x)^2 + C$
Step-by-Step Solution
Key Concept: Use the sign of the discriminant and leading coefficient to determine the appropriate substitution for integrals involving nested radicals of quadratic expressions.
Given $f(x) = \lim_{n \to \infty} \frac{x^{2n}-1}{x^{2n}+1} = -1$ for $0 \le x 0$, make the substitution $\sqrt{9x^2 + 4x + 6} = u \pm 3x$ to rationalize the radical. This converts the integrand into a form amenable to standard techniques, allowing the integral to be evaluated in closed form.
Correct Answer: 1