Problem Sets
Published:
Last Update: 2026-01-02
Remark:
$\lim_{x \to \infty} \left(1 - \frac{2}{x}\right)^{x}$
$\lim_{x \to 0} \dfrac{\int_{\cos x}^{1} e^{-t^{2}} dt}{\sin^{2} x}$
For what values of $a$ and $b$ is \(\lim_{x \to 0} \left( \frac{\tan 2x}{x^3} + \frac{a}{x^2} + \frac{\sin bx}{x} \right) = 0?\)
Suppose that $f(x)$ is differentiable on $[0,1]$, and $f(0) = 0$. Show that \(\lim_{x \to 0^+} x^{f(x)} = 1.\)
$\displaystyle \lim_{x \to 0} \frac{e^{3x}(2x\cos x - 2x + x^{3})}{\sin^{5} x}$
$\displaystyle \lim_{x \to \pi/2} \frac{\tan x}{\tan 5x}$
$\int \sqrt{1 - \sqrt{1 - x}} \, dx$
$\displaystyle \int \frac{x^{3} - 10x^{2} + 28x - 23}{(x-1)(x-2)(x-3)^{2}} dx$
$\int \sec^{3} x \, dx $
$\displaystyle \int \frac{1}{\sin^{3} x \cos x}\, dx$
$\displaystyle \int_{0}^{1} \frac{2x \sin^{-1} x}{\sqrt{1 - x^{2}}}\, dx$
$\int_{0}^{\pi} \sqrt{\cos^{2} x - \cos^{4} x} \, dx $
$\displaystyle \int \frac{1}{1 + \sqrt[3]{x + 2}} dx $
$\int x\sin^{-1} x\,dx $
$\int x^{2}\tan^{-1} x\,dx $
$\displaystyle \int \frac{\cos^{-1}\sqrt{x}}{\sqrt{x - x^2}} dx $
$\displaystyle \int \frac{\sin(\ln x)}{x^3} dx $
$\displaystyle \int \frac{1}{x(x^2+1)^2} dx $
$\int \sin^{3}x\cos^{2} x\, dx$
$\displaystyle \int \frac{x^{2}}{\sqrt{9 - x^{2}}}\,dx$
$\int x\ln x\, dx$
$\int e^{x}\cos x\, dx$
$\displaystyle \int_{1}^{\infty} 2\pi \frac{1}{x} \sqrt{1 + \frac{1}{x^4}} dx$
$\displaystyle \int_{0}^{\pi/4} \frac{\sec^{2} x}{(8+19\tan x)^{4/3}} dx $
$\displaystyle \int \frac{x^{3}+1}{x^{3} + 5x -6}\,dx$
$\int \cos(\ln x)\,dx$
$\displaystyle \int \sqrt{\frac{x}{1-x^{3}}}\,dx $
$\displaystyle \int \frac{1}{\sqrt{25x^{2}-4}}\, dx$
$\int_{0}^{\pi/4} \sqrt{1 + \cos 4x}\,dx$
- $\displaystyle \int \frac{1}{x(x^{2}+1)}\,dx$
- $\displaystyle $
$\int_{-\ln 2}^{\ln 2} x^{-2} e^{-1/x} dx $
For what value or values of $ a $ does the following integral \(\int_{1}^{\infty} \left( \frac{ax}{1 + x^{2}} - \frac{1}{2x} \right) dx\) converge? Evaluate the corresponding integral(s).
Find the value of $p$ for which the following integral \(\int_{2}^{\infty} \frac{dx}{x(\ln x)^{p}}\) converges.
Find the value of $p$ for which the following integral \(\int_{1}^{2} \frac{dx}{x(\ln x)^{p}}\) converges.
Suppose that \( x \) and \( y \) satisfy the following equation: \( x = \int_{0}^{y} \frac{1}{\sqrt{1+4t^{2}}} dt\). Show that \( \frac{d^2y}{dx^2} \) is proportional to \( y \) and find the constant of proportionality.
Assume that the function $f(x)$ is continuous, nondecreasing and nonnegative on $[0, \infty)$. Show that for any positive $\alpha \in \mathbb{R}$ \(F(x) = \begin{cases} \frac{1}{x}\int_{0}^{x} t^{\alpha} f(t)dt, & x > 0 \\ 0, & x = 0 \end{cases}\) is continuous and nondecreasing on $[0, \infty)$.
Find the linearization of $f(x) = 3 + \int_{1}^{x^2} \sec(t - 1) dt$ at $x = -1$.
Let $f(x) = \dfrac{x^{3}}{12} + \dfrac{1}{x}$.
Find the length of the curve determined by $f$ from $x = 1$ to $x = 4$.
Find the area of surface generated by revolving the above curve $f(x), 1 \leq x \leq 4$, about the $x$-axis.
Find the arc length of the graph of $f(x) = \dfrac{x^{3}}{12} + \dfrac{1}{x}, 1 \leq x \leq 4.$
Find the area of the surface generated by revolving the curve $y = 2\sqrt{x}, 1 \leq x \leq 2,$ about the $x$-axis.
Find the area of the surface generated by revolving the curve $y = \cos x$, $-\pi/2 \leq x \leq \pi/2$, about the $x$-axis.
Find the area of the region enclosed by the curves $y = x^{2}$ and $y = \sqrt{x}$.
The plane region enclosed by the $x$-axis and the parabola $f(x) = 3x - x^{2}$ is revolved about the vertical line $x = -1$ to generate a solid. Find the volume of this solid.
Determine the dimensions of the rectangle of largest area that can be inscribed in an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and the sides of the rectangle are parallel to axis of the ellipse.
- Compute the volume of the solid generated by revolving the region bounded by the $x$-axis, lines $y = 1 - x$ and $y = 1 - 2x$ about $y$-axis.
- Using the washer method;
- Using the shell method.
- What is the value of the surface area of the solid of revolution traced out by revolving the curve $y = 1/x, x \geq 1,$ about the $x$-axis?
- Let $a > 0$.
- Show that \(y = \frac{1}{a}\int_{0}^{x} f(t) \sin(a(x - t))dt\) solve the following initial value problem \(\frac{d^2y}{dx^2} + a^2 y = f(x), \frac{dy}{dx} = 0 \text{ and } y = 0 \text{ when } x = 0.\)
- If $f(x) = 0$, find the general solution of the above differential equation.
Let $f’\left(\sin^{2} x\right) = \cos 2x + \tan^{2} x$, find $f(x)$.
- Let $ f(x) $ be a differentiable function on $ (-1,1) $ and satisfy the following two properties: (1) $ \lim_{h \to 0} \frac{f(h)}{h} = 1 $; (2) $ f(x + y) = \frac{f(x)+f(y)}{1-f(x)f(y)} $ for all $x,y, x + y \in (-1,1) $.
- Prove that $ f(0) = 0 $;
- Prove that $ f’(x) = 1 + [f(x)]^{2} $;
- Find $ f(x) $ on $ (-1,1)$.
Solve the differential equation $\dfrac{dy}{dx} = 2x\sqrt{1 - y^{2}}, -1 < y < 1$.
Find the general solution of the following equation: \(4y''+4y'+y=0.\)
- Solve the following second order linear differential equation: \(y'' - 4y' + 4y = e^{2x}.\)