For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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1
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1answer
78 views

Limit $\lim_{t\to\infty}\frac{10t^3+3t-18}{6-kt^3} = 2$

I am working on the limit problem: $\displaystyle\lim_{t\to\infty}\dfrac{10t^3+3t-18}{6-kt^3} = 2$ so far I believe I have broken down the numerator correctly by doing: ...
3
votes
0answers
48 views

Partial fraction decomposition help!

I've been stuck on this problem for 2 hours and I have no idea where I'm going wrong. Here is the original equation: $$\int_{0}^{1}\frac{x-4}{x^2-5x+6}dx$$ I first factored and rewrote the fraction ...
1
vote
1answer
44 views

How to draw the graph of this function?

I have to draw the graph of $f(a)=\int_{-\infty}^\infty e^{-ax^2}dx$ on $(0, \infty)$. I know the graph of $g(x)=e^{-ax^2}$, which is but I don't know how to graph the integral. Thank you for ...
46
votes
4answers
2k views

An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$

I need your help with this integral: $$\mathcal{K}(p)=\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx,$$ where $\operatorname{Ai}$, $\operatorname{Bi}$ are Airy ...
0
votes
1answer
20 views

Finding the monotonicity and boundness of a series

How would you find the boundness of the following. $\frac{\sqrt{n+1}}{\sqrt{n}}$ I did $\frac{\sqrt{n+1}}{\sqrt{n}}$>$\frac{\sqrt{n+1+1}}{\sqrt{n+1}}$ $\sqrt{2}>\frac{\sqrt{3}}{\sqrt{2}}$ making ...
2
votes
2answers
84 views

Finding formula for a series

I am not sure how to find a formula for the following series. $\frac{1}{2},\frac{3}{4},\frac{7}{8},\frac{15}{16},\frac{31}{32}$ I am thinking it might be $\frac{2N-1}{N^2}$ for the numerator but ...
0
votes
2answers
37 views

Question about the notation or idea used in this proof

Could someone please explain the idea behind the red arrow. Not sure if I follow. To me it looks like treating an indefinite integral as a definite integral or am I barking up the wrong acyclic ...
1
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2answers
747 views

Proof of the limit laws (Analysis)

Hi everyone I'd like to know if my arguments for the next proof are sound or needs some changes to be correct. I hope they are not a little flaws. Proposition (limit laws): Let $(a_n)_{n=m}^\infty$ ...
5
votes
4answers
532 views

Integrate $\int x \sqrt{2 - \sqrt{1-x^2}}dx $

it seems that integration by parts with some relation to substitution... $$ \int x \sqrt{2-\sqrt{1-x^2}} = \frac{2}{5} \sqrt{2-\sqrt{1-x^2}} \cdot \sqrt{1-x^2}+\frac{8}{15}\sqrt{2-\sqrt{1-x^2}}+c ...
0
votes
1answer
105 views

Integration: $\int_{}\frac1{(x^2-1)^2}\mathrm dx$

I am having trouble integrating the following integral: $$\int_{}\frac1{(x^2-1)^2}\mathrm dx$$
8
votes
1answer
205 views

Closed form of $\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$

Could you please help me to solve this integration problem? $$\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$$ Its approximate numeric value is $0.419197813818367...$, but I could not find an exact ...
2
votes
1answer
67 views

Proving convergence Using the definition [duplicate]

Use the definition of convergence to prove if $x_n$ converges to $5$, then $\frac{x_n+1}{\sqrt{x_n-1}}$ converges to $3$.
3
votes
2answers
289 views

Integrating $\int \frac{x^2}{\sqrt{2+3x}}$ using u-sub and integration by parts

I'm supposed to integrate this problem by two methods and show that they are the same or that they differ by a constant. Problem: $$\int \frac{x^2}{\sqrt{2+3x}} dx$$ First method: u-sub with $u = ...
0
votes
1answer
40 views

got two different answers for a derivative.

The homework problem. I have the answer sheet $ f\left( x \right) = \left(x-1 \right)^2 \left(x+3\right)$ $f^{\prime}\left(x\right) = 2\left(x-1\right)\left(x+3\right) + \left(x-1\right)^2$ I ...
1
vote
3answers
48 views

Integrating Another Function assistance required

Integrate $$\int \frac{1}{\sin^4(x)\cos^4(x)}dx$$. So I know that for this one we have to use a identity or u substitution, integration by parts in probably not going to help....can someone please ...
1
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2answers
107 views

How to calculate the derivate of this function?

I have to calculate the derivate of this function: $$f(x)=\int_0^x g(s,x)ds $$ They don't specify what is $g$ but it's just another function. I think I could use the Fundamental Theorem of Calculus ...
2
votes
1answer
41 views

How to start showing that a formula is valid

This is the problem on the practice exam for preparing the exam, this question seems impossible to solve, how to I get started guys? Appreciated for the tutoring in advance! ...
2
votes
1answer
56 views

Evaluate integral trouble

I'm having trouble evaluating this integral. I tried $u$-substitution and integration by parts but they didn't work. Any ideas guys? Evaluate $$\int \frac{\ln(\sin x)}{\sin^2 x} dx$$ Thanks you ...
2
votes
4answers
366 views

Showing that an unbounded set has an unbounded sequence

Suppose I have an unbounded set $S$ of Real numbers. I want to show that I can find a sequence $\{a_n\}$ in $S$ such that $ \lim_{n \rightarrow \infty}a_n=\infty$. Here is what I have so far: Since ...
0
votes
1answer
63 views

Operations between Subsequences' proof

Let $A$ and $B$ be non-empty bounded subset of $R$ a) If $C={x+y:x∈A,y∈B}$, prove that $C$ is bounded above and that $\sup ⁡C=\sup⁡ A+\sup ⁡B$. b) If$ D={x-y:x∈A,y∈B}$, prove that $D$ is ...
0
votes
1answer
59 views

Consequence of completeness axiom proof

Prove that , for a set $A⊂R$, $s=sup⁡A$ if and only if i) $a≤s$ for all $a∈A$. ii) For any $ε>0$, there exists $a∈A$ such that $a>s-ε$ I can easily prove that if $s=sup⁡A$ then $a≤s$ for all ...
1
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2answers
316 views

Find the integration of $\sec(x)$ and prove it

My hw told me to prove the integral of $\sin(x)$, $\cos(x)$, $\tan(x)$, but when I get to $\sec(x)$ I'm stuck. I can find a way to prove it. Please help on explaining the integral of $\sec(x)$. ...
1
vote
2answers
112 views

Assistance evaluating $\int_0^{2\pi}\frac{1}{2}\left(2 \sin \theta + \cos \theta\right)d\theta$

I need help with evaluating the following integral $$\int_0^{2\pi}\frac{1}{2}\left(2 \sin \theta + \cos \theta\right)d\theta$$ I have attempted this but I am not sure how to complete the problem.
3
votes
2answers
85 views

Number of real roots of the equation $2^x = 1+x^2$

Find the number of real roots of the equation $2^x = 1+x^2$ My try: Let we take $f(x) = 2^x-1-x^2$. Now for Drawing Graph of given function, we use Derivative Test. $f'(x) = 2^x \cdot \ln ...
10
votes
1answer
119 views

Find asymptotic of recurrence sequence

Given a sequence $x_1=\frac{1}{2}$, $x_{n+1}=x_n-x_n^2$. It's easy to see that it limits to $0$. The question is: is there exists an $\alpha$ such, that $\lim\limits_{n\to\infty}n^\alpha x_n\neq0$. ...
0
votes
2answers
52 views

Volume using the shell method

The shell radius is $x+4$ and shell height is $\sqrt{x}-x$, the thickness is $dx$. I have determined that the boundaries are from $0$ to $1$. $$V=2\pi\int_0^{1}(x+4)(\sqrt{x}-x) dx$$ I simplified ...
4
votes
1answer
361 views

Hilarious Comic … DiffyQ and infinity ensue…

I ran across this comic, and it's gold. It is orginially published here If I am correct, the first panel alone defines a self-referential loop if not a differential Equation: $X$: Amount of Black ...
3
votes
3answers
86 views

Find $f'(0)$ for $f(x)=(2x+1)^3(3x+3)^2$.

Find $f'(0)$ for $f(x)=(2x+1)^3(3x+3)^2$. Do I use the chain rule for each or do I use the derivative product rule first Please Help!!!
2
votes
2answers
81 views

Evaluating a trigonometric function

Is it possible to find real $x$, $y$, and $s$, with $s \in (0,1)$, such that $$\cos\left(\frac{x - y}{2}\right)\sin\left(\frac{x + y}{2}\right)\cos\left(sx + (1 - s)y\right)$$ equals $1$? The thing I ...
0
votes
1answer
75 views

Project Based Calculus II

![How do I do this problem? I thought this was the correct answer THoughts? Thanks guys!]2
0
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2answers
47 views

Find the volume between $y = 4 − \frac{3x}{2}$ and $y=0$ and $x\in [0, 1]$

Find the volume $V$ of the solid obtained by rotating the region bounded by the given curves about the specified line. $$y = 4 − \dfrac{3}{2x},\ y = 0,\ x = 0,\ x = 1$$ about the $x$-axis. I ...
1
vote
3answers
72 views

Determine $f'(12)$ when $f(x)=x\cos(12/x)$. Chain Rule

Determine $f'(12)$ when $f(x)=x\cos(12/x)$ So I'm learning about chain-rule. I know that $x\cos(x)$ is the outside and $(12/x)$ is the inside. So I did derivatives for both sides. So my function is ...
2
votes
2answers
297 views

Origin of the Power Rule Proof: Who first proved the power rule?

Who was the first person to prove the power rule for derivatives? The person could have proved the power rule using limits and the binomial theorem or difference of two nth powers, or the implicit ...
2
votes
2answers
162 views

Proof of $X+\csc{\left(\frac{\pi}{X}\right)}>2\csc{\left(\frac{\pi}{2X}\right)}$

When $X$ is greater than 1, I want to prove that $X+\csc{\left(\frac{\pi}{X}\right)}>2\csc{\left(\frac{\pi}{2X}\right)}$ where $\csc{(\cdot)}=\frac{1}{\sin{(\cdot)}}$. Plotting the above ...
0
votes
2answers
250 views

Finding integral of functions involving e raised to another function.

$$\int_0^{\pi/4}(1+e^{\tan\theta})\sec^2\theta\, d\theta$$ I have tried to let $u=\sec^2\theta$ so that $du=\tan\theta \, d\theta$. After doing that I was unable to figure a way to substitute $u$ and ...
2
votes
2answers
148 views

Integral of natural log function using substitution

$$\int_{2} ^{4}\dfrac{dx}{x(\ln x)^2}.$$ Here is what I did: $$u=\ln x, du=\dfrac{dx}{x}$$ $$\int_{2} ^{4}u^{-2}du$$ $$(-1)u^{-1} |_{2}^{4}$$ $$-\dfrac{1}{\ln x}|_{2}^{4}$$ $$-\dfrac{1}{\ln ...
0
votes
2answers
375 views

Magnitude of 3D vector functions in relation to its derivative

If the magnitude of a 3D vector function is always 1, then is the derivative of that function always perpendicular to the original function?
0
votes
1answer
167 views

Function of ellipse in the first quadrant correct?

If y = (b/a)*sqrt(a^2-x^2) is the top half of the ellipse, shouldn't the function for the ellipse in the first quadrant be half the area of the top half of the ellipse? Why is it 1/4 of the area if ...
3
votes
2answers
409 views

What is the distance between the line and plane if it is parallel?

So far, I've gotten that the line is parallel to the plane $x = 2 + t$, $y = -3 + 2t$, $z = 1 + 4t$ With the vector of that being $U$ is $(1,2,4)$ and the plane $2y-z = 1$ with the vector $V$ being ...
1
vote
0answers
48 views

Solving a system of partial derivatives.

$uu_{x} - vv_{y} = 0$ $uu_{y} + vv_{x} = 0$ The subscripts represent partial derivatives. In general, the solution to this system should just be $0$. Not sure how to get that though. I was playing ...
3
votes
2answers
101 views

Calc 101 Question on simplifying a fraction

$$\lim_{h \to 0} \left(\frac 1h -\dfrac{1}{h^2+h} \right).$$ What do I do about the denominators?
24
votes
1answer
459 views

Integral $\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$

I encountered this scary integral $$\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$$ where $_3F_2$ is a generalized hypergeometric function ...
1
vote
3answers
286 views

Proof for limit superior's property: $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ [duplicate]

Let $a_n,b_n>0$ for all $n\in\mathbb N$. Prove that $\limsup (a_n b_n ) \leq \limsup a_n \cdot \limsup b_n$ I know that $\limsup (a_n+b_n ) \leq \limsup a_n + \limsup b_n$. But I don't know how ...
0
votes
2answers
92 views

Monotone subsequence proof

prove that " Prove that every sequence a_n has a monotone sub-sequence." I tried to prove this by a proof of contradiction, so I assume there exist a sequence that doesn't have monotone sub-sequence, ...
4
votes
2answers
254 views

limit superior and limit inferior proof

$$\limsup \left(\frac 1{a_n} \right)=\frac 1{\liminf(a_n )} $$ I know this is true base on the definition of $\limsup$ and $\liminf$, but I don't know how to prove it formally.
0
votes
1answer
509 views

Find the inverse function of $ f(v) = \frac{m_0}{\sqrt{1 − {v^2/c^2}}}$ and explain its meaning.

In the theory of relativity, the mass of a particle with speed $v$ $$m = f(v) = \frac{m_0}{\sqrt{1 − {v^2/c^2}}}$$ where $m_0$ is the rest mass of the particle and $c$ is the speed of light in a ...
0
votes
1answer
48 views

Solving or factors the given polynomial.

I have a polynomial and would like to solve it for "r". We can also do factorization if possible but important thing is to find the values of r. We will get possibly three solutions from this ...
0
votes
2answers
493 views

Integrate $\int \frac{dx}{(4x^2-9)^{3/2}}$

How to solve the integral $$\int \frac{dx}{(4x^2-9)^{3/2}}$$
3
votes
3answers
138 views

Integrate $\int \frac{x^2}{(x^2+4)^2}$

$$\int \frac{x^2\;dx}{(x^2+4)^2}$$I suppose you must have to use trigonometric substitution, or something, but do not even know where to begin to solve this integral, guys, please help me!!
1
vote
4answers
92 views

integrating $\int \frac{dt}{(t+2)^2(t+1)}$

I'm practicing to solve a whole, and I am not able to solve this one, could you help me? $$\int \frac{dt}{(t+2)^2(t+1)}$$I tried ...