Integral Calculus-1
Integral Calculus-1
Allen Star Batch
Grade 12
Question:
If $f(x) = \lim_{n \to \infty} \frac{\tan(1/n)\log(1/n)}{n}$, and $\int \frac{f(x)}{\sqrt{\sin^{11} x \cos x}} dx = g(x) + C$ (C being the constant of integration). Then:
$g\left(\frac{\pi}{4}\right) = \frac{3}{2}$
$g(x)$ is continuous for all $x$
$g\left(\frac{\pi}{4}\right) = \frac{15}{8}$
$g\left(\frac{\pi}{4}\right) = -\frac{1}{2}$
Step-by-Step Solution
Key Concept: Recognize that $\lim_{n \to \infty} n \cdot \tan(1/n) \log(1/n) = 0$ using $\tan u \sim u$ and logarithmic growth rates, making $f(x) = 1$.
We find $f(x) = \lim_{n \to \infty} e^{\tan(1/n) \log(1/n)}$ by evaluating $\lim_{n \to \infty} \tan(1/n) \log(1/n) = 0$, so $f(x) = e^0 = 1$. The integral $\int \frac{f(x)}{\sqrt[3]{\sin^{11}x \cos x}} dx$ is solved via substitution $t = \tan x$, yielding $\int \left(t^{-11/3} + t^{-5/3}\right) dt = \frac{3}{8}t^{-8/3} - \frac{3}{2}t^{-2/3} + C = \frac{3}{8} \frac{(1+4\tan^2 x)}{\tan^2 x \sqrt[3]{\tan^2 x}} + C$.
Correct Answer: 3