11
votes
3answers
217 views

Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...
12
votes
1answer
139 views

$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$ [duplicate]

$$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$$ I tried Integration by parts but failed. Wolfram alpha gives answer in decimal points which are same as of $2\pi$. Any hints or suggestions will be helpful.
2
votes
1answer
30 views

Check the properties of the eigenfunction corresponding to the distinct eigenvalues of an integral equation

Let $\lambda_1, \lambda_2$ be eigenvalues and $f_1 , f_2$ be the corresponding eigenfunctions for the homogeneous integral equation \begin{align} \phi(x) - \lambda \int_0^1 (xt +2x^2) \phi(t) ...
3
votes
5answers
192 views

Calculating the area

For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them; Attempt to solve: Limits of the integral are $1$ and $-3$, so I took ...
6
votes
3answers
195 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
-1
votes
1answer
42 views

Value of line integral [on hold]

Let $C=\{(x,y)\in \Bbb R^2:~\max\{|x|,|y|\}=1\}.$ The value of the line integral $$\oint_C(xy^2+2y+sin(e^x))dx+(x^2y+cos(e^y))dy$$ is?
1
vote
1answer
40 views

Volumes of Revolution Washer Method

I have to find the volume of revolution of a region called $C$ using around the $y=-1$ axis. The region is bounded above by $y \ = \ \ln(x+1)$, bounded below by $y=e^{-x}$ and on the right by $x=3$. ...
1
vote
1answer
38 views

Arc Length in two dimensions by integration

I'm really at the end of my wits on this problem. Basically I'm trying to find arc length. The vector-valued function is: $R=\langle t,\sqrt{t}\rangle$ and $t\ge0$. We're looking for the length of ...
2
votes
3answers
101 views

How to calculate integral $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$?

The integral is $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$. I used Mathematica to calculate, the result was $\dfrac{2\pi}{\sqrt{3}}$, I think it may help.
-2
votes
2answers
40 views

Fourier transform of t*(sent/pi*t)^2

Here's the function (I need it's fourier transform).
1
vote
1answer
48 views

Solve the integeral equation (C.S.I.R)

Let $\lambda_1, \lambda_2$ be the eigen value and $f_1 , f_2$ be the coressponding eigen functions for the homogeneous integeral equation $$ \phi(x) - \lambda \int_0^1 (xt +2x^2) \phi(t) dt = 0 $$ ...
3
votes
3answers
132 views

How do I evaluate the integral $\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$?

I have no idea how to start, it looks like integration by parts won't work. $$\int_0^{\infty}\frac{x^5\sin(x)}{(1+x^2)^3}dx$$ If someone could shed some light on this I'd be very thankful.
0
votes
1answer
67 views

physics math problem with dirac delta

I'm trying to do a homework problem where they have asked me to recover the usual coordinate space momentum operator from its hilbert space equivalent. it gets you to show: ...
0
votes
2answers
40 views

How to evaluate this triple integral?

How would I go about evaluating this integral? I want to change the order of integration but don't know how. $$\int_0^1\int_1^{\Large e^z}\int_0^{\log y}x\ dx\,dy\,dz$$ I'm having difficulty ...
0
votes
3answers
58 views

Evaluate integral

How do I evaluate the following integral, the answer according to Wolfram Alpha is $2$, but I keep on getting $0$ after using integration by parts.$$\frac12\int_{-\infty}^\infty x^2e^{-|x|}\ dx$$
1
vote
0answers
43 views

How does one calculate $\int_0^1 \frac {\arcsin(x)}{x}dx$? [duplicate]

How can I evaluate the following? $$\int_0^1 \frac {\arcsin(x)}{x}dx.$$ Could not find a primitive, so I went for some other methods like arranging it as a double integral or introducing a ...
0
votes
1answer
12 views

A question about Integrability and Uniform Continuity

I got this questions: Prove or disprove by a counterexample the following statements: Let $f:\mathbb{R}\to\mathbb{R}$ be a function that is integrable on every closed interval and let $F(x)=\int_0^x ...
0
votes
1answer
35 views

Some questions about integrals

I got those questions: Prove or disprove the statement by a counterexample: (1) If $f$ is a continuous function on $[a,b]$, Then there exist $c\in(a,b)$ such that $\int_a^b f(x)\;dx = f(c)(b-a)$. ...
0
votes
2answers
44 views

Integrability of a piecewise function [closed]

I got this question: Is the function $g$ defined below is integrable on $[0,1]$? $$g(x)= \begin{cases} \dfrac{1}{x}\sin\dfrac{1}{x} & \text{if $x\neq 0$} \\[8pt] 3 & \text{if $x=0$} ...
1
vote
2answers
47 views

If $\int_a^b f(x)\;dx=(b-a)\sup\;f([a,b])$ where $f$ is continuous on $[a,b]$ then $f$ is constant in $[a,b]$

I got this question: Let $f$ be a continuous function on $[a,b]$ that satisfies $\int_a^b f(x)\;dx=(b-a)\sup\;f([a,b])$, Must it be the case that $f$ is constant in $[a,b]$ ? I tried to find a ...
1
vote
1answer
18 views

If the upper Darboux integral of a non-negative bounded function $f$ equals zero then $f$ is integrable

I got this question: Let $f$ be a non-negative, bounded function on the interval $[0,1]$ such that it's upper Darboux integral equals 0, i.e. $\int_0^{1^-} f(x)\;dx = 0$. Prove or disprove by a ...
8
votes
6answers
234 views

$\int_{0}^{+\infty} \frac{1-\cos t}{t} e^{-t}dt=?$

I have a question: $$\int_{0}^{\infty} \frac{1-\cos t}{t} e^{-t}dt=\ ?$$ Thanks for your any help. Thanks ahead.
0
votes
0answers
25 views

convergence of integral with two parameters

for which values of $p$ and $q$ the following integral is converges? $$\int_{0}^{1} \frac{(x-x^5)^p}{\sin^q{(x)}}$$ the correct answer is: $q<p+1$, $p>-1$. my solution is: since $x=0$ ...
0
votes
2answers
34 views

Integral Test for Convergence - Log of a Log

I need to investigate the convergence of the series $\sum_{n=3}^{\infty}\dfrac{1}{n\ln n}$ So, doing the integral test, I end up with (shortcutted because integration is boring): ...
0
votes
2answers
33 views

Definite Integral Evaluation

Evaluate $\displaystyle \int_{0}^{2}x^3\sqrt{(2x-x^2)} dx$ This kind of problem is solved using tricks like putting $\displaystyle x=\sin^2t$ and/or identities like $$\begin{align} ...
3
votes
2answers
77 views

Find integral $\int_0^\infty\frac{dx}{(x^3+(1+x^2)^{3/2}+x)(\sqrt{1+x^2} \tan^{-1} x+1)}$

How to find this integral $$\int_0^\infty\frac{dx}{(x^3+(1+x^2)^{3/2}+x)(\sqrt{1+x^2} \tan^{-1} x+1)}$$ This one is very difficult for me because I missed my calc class twice and I didn't know about ...
3
votes
1answer
52 views

Calculate integral using the method of parameter derivation or integration [duplicate]

$$F( \alpha ) = \int^{ \pi }_{ 0} \ln ( 1 - 2 \alpha \cos x + \alpha^{2} ) dx$$ Should I derive the inner function? But I can't process the derived outcome.
1
vote
1answer
50 views

Evaluate double integral:

Evaluate: $$\iint_{D} \arcsin(x^2+y^2)\, dx\,dy$$ where $D$ is defined by the following polar equation $\rho=\sqrt{\sin \theta}$ and $0\le\theta\le\pi$
5
votes
8answers
213 views

Evaluate $\int_0^1\frac{x\ln x}{(1+x^2)^2}\ dx$

$$\int_0^1\frac{x\ln x}{(1+x^2)^2}\ dx $$ Help me please. I don't know any ways of solution. Thank you.
4
votes
1answer
41 views

Finding the area under a curve represented by the equations $x=a\cos{t}+\frac{a}{2}\ln{\left(\tan^2{\frac{t}{2}}\right)}$ and $y=a\sin t$

How do I find the area of the curve represented by the following equations, $$x=a\cos{t}+\frac{a}{2}\ln{\left(\tan^2{\frac{t}{2}}\right)}\\ y=a\sin t$$ Here's what I tried: Let $A$ denote the area ...
5
votes
2answers
96 views

Find the value of the integral $\int_0^{2\pi}\ln|a+b\sin x|dx$ where $0\lt a\lt b$

Find the value of the integral $$\int_0^{2\pi}\ln|a+b\sin x|dx$$ where $0\lt a\lt b$. What is the use of this inequality. I tried to integrate the integral by parts, but the integral of the 2nd term ...
1
vote
1answer
34 views

Integration by Substitution question

I just wanted to check if I did this question correctly or if I made a mistake when calculating $\frac{du}{dx}$
2
votes
1answer
75 views

Line integral answer confirmation please :). I have moved the actual question to the first line.

My question: Am I meant to sub in something for $x$ and $y$ below? I believe I have now obtained the correct answer: $\oint_C \mathrm{F\cdot T \;ds} \;= 4xy + 4x^2 - 4xy - 4x^2 = 0 $ The ...
1
vote
2answers
64 views

Approximate a definite integral to three decimal places: $\int_0^2 \frac{dx}{\sqrt[3]{64+x^3}}$.

I try to expand function $$\frac1{\sqrt[3]{64+x^3}}$$ using Maclaurin series. So, $f(x) = 64{(1+ \frac{x^3}{64})}^{-1/3}$. I expand it and I get ...
1
vote
1answer
25 views

Triple integral basic

Need some help on how to think about this question: $$\iiint_K(x^2+y^2)dxdydz$$ $$0\leq x^2 + y^2 \leq z^2\\ 0\leq z \leq 1$$ I have been using spherical solution but does not really get the right ...
1
vote
4answers
52 views

How to calculate this limit involving an integral

I missed a couple of classes so I'm having trouble doing this (and other similar) excercises of homework: $\lim\limits_{x\to0^+}\frac{\displaystyle\sqrt{x}-\displaystyle\int_0^\sqrt{x} ...
1
vote
2answers
35 views

Find the area of the surface of revolution generated by revolving about the $x$-axis the hypocycloid $x=a\cos^3\theta$, $y=a\sin^3\theta$

Find the area of the surface of revolution generated by revolving about the $x$-axis the hypocycloid $x=a\cos^3\theta$, $y=a\sin^3\theta$ ($0 \leq \theta \leq \pi$) I know you have to integrate $2\pi ...
0
votes
1answer
51 views

Double integral of $\ln(x^2+y^2+1)$

I'm have some trouble with this one, maybe someone can help. Domain: (1 $\leq$ $x^2 + y^2 $ $\leq$ 2) $$\iint_D \ln(1+x^2 + y^2)dxdy$$ I have a hard time getting the right answer: $\pi( ...
0
votes
2answers
101 views

Help my homework: $\int_0^1\int_{0}^1\frac{x^2-y^2}{(x^2+y^2)^2}dy\, dx$ [duplicate]

I am trying to integrate $$\int_0^1\int_{0}^1\frac{x^2-y^2}{(x^2+y^2)^2}dy\, dx$$ In my book said that use tangent function but I don`t know how to evaluate it. Please help me. I want to know the ...
5
votes
2answers
82 views

Calculate $\int_0^1 e^x dx$ as a limit of a sum?

As for now, I've been doing the opposite thing. For a given sum in terms of $n\in\mathbb{N}$ I had to calculate the limit (as $n$ approaches infinity) of that sum by applying: ...
4
votes
3answers
80 views

Evaluate Gauss-like Integral

Evaluate Integral $$\int_0^\infty e^{-ay^{2}-\frac{b}{y^2}}dy $$ Where a and b are real and positive. This integral is eerily similar to the Gaussian integral $$\int_0^\infty e^{-\alpha x^2}dx = ...
0
votes
1answer
49 views

Polar Coordinates

I dont really know how to go about this question. I know that the area is $$\int_\alpha^\beta \frac12 r^2\, d\theta $$ The question is to find the area of the shaded region.
1
vote
1answer
76 views

Calculate $\lim_{n\rightarrow\infty}\frac{1}{n}\left(\prod_{k=1}^{n}\left(n+3k-1\right)\right)^{\frac{1}{n}}$

I'm need of some assistance regarding a homework question: "calculate the following: $\lim_{n\rightarrow\infty}\frac{1}{n}\left(\prod_{k=1}^{n}\left(n+3k-1\right)\right)^{\frac{1}{n}}$" Alright so ...
1
vote
4answers
234 views

Can someone help me to evaluate this integral:

Can someone help me with evaluating this integral: $$\int_{1}^{2} \frac{2x^2-1} {\sqrt{x^2-1}}\, dx$$ I tried using integration by parts, integration by substitution....but nothing...
1
vote
0answers
49 views

Prove the following: $\int_{a}^{b}|f\left(t\right)|dt\leq\left(b-a\right)\int_{a}^{b}|f'\left(t\right)|dt$

I have this homework question and I'm in need of some assistance: "Let there be a function $f:\left[a,b\right] \rightarrow\mathbb{R}$ continuously derivatable (every derivative is continuous), and ...
2
votes
1answer
52 views

Triple integral in spherical coordinates

I'm trying to evaluate the triple integral $\int\int\int_B\frac{dV}{\sqrt{x^2+y^2+z^2+3}}$, where $B$ is the ball of radius $2$ centered at the origin. Both the integrand and the nature of $B$ ...
6
votes
2answers
217 views

Definite integral problem

I was solving a definite integral problem which was reduced to : $$\int^{1}_{0} \frac{\ln(1+t)}{t} dt$$ I couldn't solve it and when I saw the solution, the answer was simply given as ...
2
votes
0answers
31 views

Integral Evaluation: Exponential of and Hyperbolic Function

I'm trying to evaluate $$G^{\pm} = \frac{-i}{8\pi^2 X} \partial_X \int_{-\infty}^\infty d\phi e^{i m \left[X \sinh \phi \pm T \cosh \phi \right]}$$ for $T = \pm X$. Where $T, X, m \in \mathbb{R}$ ...
2
votes
2answers
68 views

Problem calculating line integral

I have $\gamma=[0,1]\to\mathbb{R}^3$ defined by $\gamma(t)=(\cos(2\pi t), \sin (2\pi t), t^2-t)\;\forall t\in[0,1]$ and I'm asked to calculate ...
0
votes
1answer
36 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!