Calculus

I would like to prove that $\int_0^{\frac{\pi}{2}}\frac{ x}{ \tan x}dx=\frac{\pi}{2} \ln 2.$ None of the integration techniques seems to work.

$\int_{0}^{1} \frac{dx}{x-sinx} $Hi. Instructions for this problem are to evaluate the improper integral or show that it diverges. I'm stuck at integrating this function. If someone could please expla...

$\int_{-\infty }^{0} xe^x dx$Hi. Online calculators are telling me this converges to -1. My first instinct is that it diverges to negative infinity but I get what looks like an indeterminate form when...

Find the volume of the solid obtained by rotating the region bounded by $f(x)=\sin x$, $y=0$, between $x=0$ and $x=\pi$ about $x$-axis.

Attached file has 4 questions to answer. I have attempted these but would appreciate seeing someone else attempt them before I submit. Must show all working please.

Hey, I really liked your response to the last question I asked. Could you please explain what the person means by "vanishing determinant condition," and how he derives that $h=2\sqrt{(2r+R)(r+2R)} $ a...

Hello guys, quick question. I'm attempting to optimize the surface area of a frustum but I'm having trouble understanding how to get from one step to another. Could anyone please explain why this pers...

I am making something on desmos and I need the maximum gradient of a function within a domain. I have no idea where to start.f(x) {x:a<=x<=b}find f'(x) max The desmos graph plots the function f(...

7 Let f : R 7→ R be a function with sup x∈R |f(x)| ≤ Bfor a finite number B. Show by an ε − δ argument that the function F : R 7→ R given byF(x) = integral from 0-x f(s)ds is continuous at each point...

3 Show, using $\epsilon-\delta$ definitoon of continuity, that $f : x → x^3$ is a) continuous at each fixed point $x_0 \in \mathbb{R}$,b) unifonrmly continuous on $[0,a]$.

Attached file has 4 questions to answer. I have attempted these but would appreciate seeing someone else attempt them before I submit. Must show all working please.

6. Consider the function $f(x) = (e^{−2x})/(2x)$ defined in all of R. Find and classify the extremal values of f. Does f have any asymptotes? If so, find them.

1. For each expression, calculate the limit if it exists. If the expression does not converge, decide if the expression tends to +∞ or −∞.• limx→0  ((sinx)^2)/x^n) for n = 0, 1, 2, . . .

4. Consider the function f : R → R, f(x) = ⌊x⌋, given by discarding the decimalpart of x: f(x) = max {y ∈ Z | y ≤ x}. For example, ⌊5.8⌋ = 5 and ⌊−2.3⌋ =−3. Find all points at which f is not continuou...

2. For f(x) = (lnx)/x , x > 0, compute the Taylor polynomial P(x) of degree three centered at the point x = e. Using the Lagrange form of the remainder term, find an interval (e−δ, e+δ) in which th...

A cylindrical tin can is to be manufactured so that it will hold a specific volume V. If the materials for the ends of the can are twice as expensive as materials for the sides, what shape of can is m...

A telephone line hangs between two poles 14 meters apart in the shape of a catenary $y = 20cosh(\frac{x}{20})-15 $ , where x and y are measured in meters. Find the length of telephone wire needed betw...

John wants to get to the bus stop, the bus stop is across a grassy park, 2000 feet west and 600 feet north of his starting location. John can wal west along the edge of the park on the sidewalk at a s...

I've attached my question as a file and here is the resource I was looking at https://cedar.wwu.edu/cgi/viewcontent.cgi?article=1012&context=wwu_honors

What is the initial velocity of a rock that is thrown vertically to a height of 100m before it returns to earth? Use g = 9.8m/s^2 and note that h"(t) = -g for the height h(t)

Rewrite $\int_{\sqrt2}^{2\sqrt2} \int_{-\pi/2}^{-\pi/4} r^2cos(\theta)d\theta dr$ in cartesian coordinates (x,y).

In my Calculus class, we were solving an integration by parts problem two different ways. One way gave an answer pretty easily, the other led to an infinite series that stumped us. After playing aroun...

Determine if the following integral converge or diverge. Please show your steps.$\int_{1}^{\infty} e^{\sin(x)}\cdot\frac{\sin(x)}{x^2} $

$\int_{15}^{\infty} \frac{-6x^4-7x^2+1}{7x^4-x^3+x}dx $

$\int_{1}^{\infty}\frac{cos(e^x)+sen(e^{-x})}{e^{x}+e^{-x}}dx $