Calculus Practice Set 1

Let $y = (\sin 2x)^x (\arcsin 2x)^x$, find $\frac{dy}{dx}$.

Given $y = y(x)$ determined by $y = 3f(xy) + \ln(1-\sin x)$, find $\frac{dy}{dx}$.

Given parametric equations $\begin{cases} x = 3t^3+2t+3 \\ y = \int_0^t(3u+1)\sin u^2\,du \end{cases}$, find $\frac{d^2y}{dx^2}\bigg|_{t=t_0}$.

Calculate: $\int_1^2 \frac{1+\sqrt[3]{x}}{1.1\sqrt[3]{x^2}}\,dx$

Calculate or state: $\int_1^{+\infty}\frac{1}{x^2\sqrt{x^2-1}}\,dx$

Find: $\lim_{x\to 0}\left(\frac{1}{\ln(1+\sin x)}+\frac{1}{\ln(1-\sin x)}\right)$

Find: $\lim_{x\to 0}\frac{\tan x - x}{\sqrt{1-x^3}}$

Find: $\lim_{x\to\frac{\pi}{2}}(\sin x)^{\cos x}$

Find $\sum_{n=1}^{\infty}\frac{(x-2)^n}{n3^n}$, convergence radius and interval.

Given $f(x)=\frac{1}{x^5-2x-3}$, determine number of zeros of antiderivative and their locations.

Calculate: $\int\frac{1+x^2-x^3}{x^3(1+x^2)}\ln(1+x^2)\,dx$

Let $f(x)\in C[0,1]$ be concave. Prove: (1) There exists $\xi\in(0,1)$ with specific property; (2) If integral condition holds, uniqueness follows.

Given $f(x)$ on $[0,+\infty)$ with $f'(x)<0$ and integral equation, find $f''(x)$ and analyze curve properties.

Let $a_n=\int_0^1\tan^n x\,dx$, $n\geq 2$. (1) Calculate $a_n+a_{n+2}$ and prove inequality; (2) Prove series $\sum_{n=2}^{\infty}(-1)^na_n$ converges.