Questions about the evaluation of specific definite integrals.

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2
votes
1answer
34 views

Integrate $\int_{0}^1 (1 + 4y^2)^{1/2} dy$ [duplicate]

$$\int_{0}^1 (1 + 4y^2)^{1/2} dy$$ So, how do I integrate this without the use of trigonometrical substitution? Can anybody give me a hint? Thank you!
2
votes
1answer
55 views

Trouble solving an integral

So I have been trying to solve this equation, The given answer is, I began by using substitution to change the integral. Substituting t back in where t is taken from 0 to infinity. ...
1
vote
1answer
35 views

Understanding a particular transformation of an integral given in a proof

Using the theorem of mean values find the sign of the integral... $$\int_{0}^{2 \pi}{\sin x \over x}dx= \int_{0}^{\pi}{\sin x \over x}dx+\int_{\pi}^{2 \pi}{\sin x \over x}dx$$ Then: $[x-\pi=t ; ...
0
votes
0answers
37 views

Applying contour integration to $\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$

Is it possible to apply contour integration to find the value of following integral $$\int_{0}^{\pi}dx\frac{cos(x)}{\sqrt{x^2 + x_0^2}}$$
0
votes
0answers
31 views

Integral of $\frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}$

Does the following integral have a closed form solution? $$ \int_{0}^{\infty} \frac{\exp\left(\, -\alpha x\,\right)\, (x-x_0)} {{(x-x_0)^2+\beta^2}}{\rm d}x $$ where $\alpha$, $\beta$ and $x_0$ are ...
0
votes
1answer
22 views

Why is just $0$ extreme point? v22

We have $f:R\rightarrow R,\:f\left(x\right)=x^3-3x+2$ and we need to find extreme points for $g:R\rightarrow R\:,\:g\left(x\right)=\int _0^{x^2}\:f\left(t\right)e^tdt$. Here is all my steps: ...
0
votes
2answers
21 views

How we can prove that $a_n=\sum _{k=1}^nf\left(k\right)-\int _0^n f(t)\:dt$ is convergent?

We have $f:\left(-1,\infty \right)\:\rightarrow \:R,\:f\left(x\right)=\frac{x}{x+1}$ and we need to prove that: $a_n=\sum _{k=1}^nf\left(k\right)-\int _0^n\:f\left(x\right)dx$ is convergent.Maybe, in ...
0
votes
1answer
24 views

Evaluate $\lim _{n\to \infty }\left(1-f\left(\frac{1}{\sqrt{n}}\right)\right)\cdot \sum _{k=1}^nf\left(\frac{k}{n}\right)$

We have $f:\left[0,\frac{\pi }{2}\right]\rightarrow R,\:f\left(x\right)=cos\left(x\right)$, and we need to evaluate: $\lim _{n\to \infty ...
0
votes
1answer
37 views

Evaluating $\lim_{n\to\infty}\int_0^1x^nf(x)\,dx$. [duplicate]

Let $f$ be a continuous function on [0,1]. Evaluate $$\lim_{n\to \infty} \int_0^1 x^nf(x)dx$$ My approach : Consider $\int x^nf(x)dx = \frac{f(x)x^{n+1}}{n+1} - \frac{1}{n+1}\int x^{n+1}f(x)dx$ ...
0
votes
3answers
41 views

Evaluate the integral in terms of areas.

I understand that the first one is 4 from basically adding the squares inside the signed area, but I'm unsure on how to proceed in getting the other integrals. Any help would be appreciated, thank ...
2
votes
2answers
49 views

If $f, g \in L^p$, is it true that $\int | f g | = \int | f | \int | g |$?

Let $f,g \in L^p(0, 1), \;\; 1 < p < \infty$. In this case, is it true that $$\underset{(0, 1)}{\int} | f(x) g(x) | dx = \underset{(0, 1)}{\int} | f(x) | dx \underset{(0, 1)}{\int} | g(x) | dx? ...
1
vote
1answer
28 views

Heaviside function & Integral Limits

When considering integration, how does one use the Heaviside function in order to alter the limits of integration. For example If i have $$ \int_a^b f(x) dx $$ But want to change this integral to be ...
-4
votes
0answers
30 views

What radius circle to remove from unit circle to make golden earring?

A circular lamina of radius $x$ is removed from a circular lamina of radius $1$. If the center of gravity is at the edge of the smaller circle (along the line connecting the two centers) what is $x$? ...
7
votes
3answers
440 views

“Length” of rationals in an interval

For $x \in \mathbb{R}$, define $r(x)$ as follows: $$ r(x)= \begin{cases} 1 &\text{if $x$ is rational},\\ 0 &\text{if $x$ is irrational}. \end{cases} $$ Q. What is $\int_0^1 r(x) dx$ ? I ...
5
votes
2answers
60 views

Evaluate $\text{k}$ from the given equation

If $$ \int_{0}^{\infty} \left(\dfrac{\ln x}{1-x}\right)^{2} \mathrm{d}x + \text{k} \times \int_{0}^{1} \dfrac{\ln (1-x)}{x} \mathrm{d}x =0$$ then find the value of $\text{k}$ My ...
1
vote
1answer
62 views

Wrong integral proof

I was told by my maths teacher that the following procedure is wrong but didn't really understand why so I hope someone here can explain it to me. We want to prove that $\int_0^{π/2} ...
1
vote
2answers
30 views

Convergence of a sequence of integrals

I've tried expanding the hinted expression by using the definition from part (i) and choosing an X0 sufficiently large that |f(x)-l| < 1 but this doesn't appear to help very much at all. I've ...
1
vote
0answers
19 views

It is possible to transform this integral invovlving modified Bessel function?

Consider this integral $$\int_{-\infty}^{\infty}e^{-2ty^2}\left(I_{\frac{-1}{2}-i\,y}(x)-I_{\frac{1}{2}+i\,y}(x)\right)\,dy$$ where $I_v(x)$ is the modified Bessel function of first kind, ...
2
votes
4answers
64 views

Integral of $\tan(x)$ from $0$ to $2\pi$

I had a disputation with my friend. He said that $$ \int_0^{2\pi} \tan(x) \ dx $$ is undefined. While I admit that the Integral from $0$ to $\frac{\pi}{2}$ goes to infinity, I don't know if the ...
1
vote
1answer
32 views

Definite integral involving negative infinity

I'd really appreciate some feedback on my calculation of the definite integral below, if you could spare the time to look over my work: $$\int_{-\infty}^{1} \frac{dx}{(3-x)^3}$$ Using the reverse ...
1
vote
3answers
74 views

Evaluating an integral using Gamma function [on hold]

For $r \in (0,2)$, I would like to evaluate the integral $$\frac{2}{r} \int_0^{\infty} \frac{\sin(u)}{u^r} du.$$ The answer should be $$\frac{\pi \cdot \mathrm{cosec}{\frac{r\pi}{2}} ...
0
votes
1answer
20 views

integral of continuous function that does not represent power of $x$

in book named Partial differential equation by Eutiquio C. Young, there exist such question : Show that there is no continuous function $f$ that satisfy $x^n=\int_a^x ...
0
votes
2answers
59 views

$\int^\infty_0 e^{-\alpha x}\sin(\beta x)\,dx = \frac{B}{\alpha^2+\beta^2}$ Laplace [on hold]

$$ \int^\infty_0 \! e^{-\alpha x} \sin(\beta x)\,dx = \frac{\beta}{\alpha^2+\beta^2} $$ Can someone start this for me? I don't know where to begin.
4
votes
2answers
63 views

Solution of Integral $\int_0^\infty \frac{x^{r/\beta}e^{-\alpha x}}{\left(1-(1-\beta)e^{-x}\right)^{\alpha+1}} \, dx$

I need to solve the following integral $$\int_0^\infty \frac{x^{r/\beta}e^{-\alpha x}}{\left(1-(1-\beta)e^{-x}\right)^{\alpha+1}} \, dx$$ where $r,\alpha,\beta>0$. kindly help me to solve this ...
2
votes
1answer
34 views

positive double integrals

Suppose $U:={\iint}_{R} (x^2 + 2y^2+9) \,dx\,dy$ and $V:= \iint _R (2x^2 + 3y^2)\, dx\,dy$. Determine the integration region $R$ where $U \geq V$. Hence, find the value $U-V$ over this region. ...
2
votes
1answer
91 views

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ [duplicate]

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ So the lecturer gave this problem. I tried this really hard but couldn't succeed. It ...
0
votes
2answers
77 views

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$? [duplicate]

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$ ? where f is an bijective function and $f(a)=b,f(c)=d,$ I don't understand graph... I can't see on graph this ...
1
vote
1answer
35 views

It's possible to prove that this integral is positive?

I would like to prove that this function $$F_t(x)=\int_{-\infty}^{\infty}e^{-\frac{u^2}{2 t}}\,\cosh\left(u\right)\,K_{1/2+i\,u}\left(\frac{1}{4x}\right)\,du,\hspace{0.5cm}x,t>0$$ is positive, ...
0
votes
2answers
40 views

How to fastest approximate definite integrals

I know that a definite integral is a limit of Riemann sums. So if one wanted to estimate a definite integral (because one might not be able to find an antiderivative), then one can just take enough ...
0
votes
1answer
25 views

How do you integrate $\int _{0}^t\:\dot p(s) p(s) + p^2(s)ds$

Given $p(s)$ some single valued function How can I show that $$\int _{0}^t\:\dot p(s) p(s) + p^2(s)ds$$ has resulting in something along the line of $$\frac{p^2(s)}{2}$$ note $\dot p(s)$ signifies ...
3
votes
3answers
237 views

Integral involving Bessel functions of the first kind

I am stuck with the following integral. Does it converge? $$ \int_{0}^{\infty}\left(J_1(x)^2+J_1(x)J_1(x)^{''}\right)\text{d}x $$ According to tables I find that the first term is divergent, so I ...
1
vote
3answers
60 views

How to find $p(t)$ when $m$ varies linearly with $t$? [closed]

I have a function $p(t)$ (position and time) defined by $$p(t) = \frac{1}{2} \cdot \frac{F}{m} \cdot t^2$$ when the mass is constant. This is derived from Newtons second law and by integration of the ...
2
votes
4answers
104 views

Compute $\int _{\frac{4}{5}}^2\:f^{-1}\left(x\right)dx$

We have to compute $\int _{\frac{4}{5}}^2\:f^{-1}\left(x\right)dx$ where $f\left(x\right)=\frac{-x^3+2x^2-5x+8}{x^2+4},\:x\in \mathbb{R}$ is an bijective function. How help if we kno![enter image ...
1
vote
2answers
48 views

Help me with Integral, $\int_1^{\sqrt 3}\frac{dx}{(1+x^2)\arctan x}$ [closed]

$$\int_1^{\sqrt 3}\frac{dx}{(1+x^2)\arctan x}$$ Can you help me with this integral?
1
vote
1answer
26 views

The value of x satisfying $\int^{2[x+14]}_0\{\frac{x}{2}\}dx =\int^{\{x\}}_0[x+14]dx $ where [.] …

Problem : The value of x satisfying $\int^{2[x+14]}_0\{\frac{x}{2}\}dx =\int^{\{x\}}_0[x+14]dx $ where [.] denotes the greatest integer function and $\{.\}$ denotes the fractional part function. ...
2
votes
3answers
45 views

How to prove and evaluate an Improper Integral

How to show that this improper integral converges and how to compute its value? $$ I=\int_{0}^{\frac\pi 2}\frac{\cos(2t)}{\sqrt{\sin(2t)}}\mathrm{d}t. $$ I used that the integrated function is odd so ...
1
vote
0answers
42 views

Is there a closed form expression for the following definite integrals?

I am looking for a closed form for these two integrals $$\int_{-\infty}^{-a}\text{d}x \frac{1}{|x|}e^{-\frac{1}{2}x^2\sigma^2}e^{i k |x|}+\int_a^{\infty}\text{d}x ...
0
votes
1answer
41 views

Evaluate $\lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt$ and a big mistake in the book

We have to evaluate $\lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt$ where $f\left(t\right)=\frac{1}{\left(1+t^2\right)\left(1+t^3\right)}$. In my book they say that $\int ...
0
votes
2answers
45 views

Is this integral correctly calculated?

The problem is that I can't use wolframalpha to check this because he is worried about integration limits: I have $a>0$ and $t \in (-1,1).$ $$(1-t)^{\frac{a}{2}} \int_0^t \frac{1}{(1-x)^{a+1} }dx= ...
0
votes
1answer
27 views

How to show this integral (Error function)

I'm given this question. Show that $$\int_{0}^{0.25}\frac{1}{\sqrt{x}}e^{-x}dx=\int_{0}^{0.5}2e^{-u^2}du$$. As I know this integral is an error function. How to show? Can anyone give me some hints? ...
9
votes
5answers
178 views

Evaluate $\int_0^{1/\sqrt{3}}\sqrt{x+\sqrt{x^2+1}}\,dx$

I want to find a quick way of evaluating $$\int_0^{1/\sqrt{3}}\sqrt{x+\sqrt{x^2+1}}\,dx$$ This problem appeared on the qualifying round of MIT's 2014 Integration Bee, which leads me to think ...
0
votes
2answers
82 views

Evaluate $\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$

My question is: Evaluate $$\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$$ for $k=0,1,...,(n-1)$ and $n \in \mathbb{N}$. I've tried integration by parts but without much success. Any ...
0
votes
2answers
24 views

How to find volume of the given solid analytically?

Here is the question - I am able to visualize the solid, but how do I find its volume? I'm unable to figure out the 2D structure that when rotated, produces this solid. Please help. Edit: The ...
2
votes
1answer
71 views

What is $\int_0^1 \frac{\log(x+1)}{x^2+1}dx$ [duplicate]

It's so deceptively simple and none of the usual techniques are working. Any and all insights are welcome.
1
vote
0answers
46 views

Estimation of a certain Integral

I estimated (w.r.t. $\varepsilon$) the expression \begin{align} &\left|\int_{-1}^{x_0-\varepsilon} (1-x)^{n-p}(1+x)^p+\int_{x_0+\varepsilon}^1 (1-x)^{n-p}(1+x)^p \, dx \right | \\[6pt] \leqslant ...
0
votes
1answer
33 views

The Gherkin (an egg shaped building) - equation for the curve in order to calculate the surface area of revolution

I am trying to calculate the surface area of revolution for The Gherkin, an egg-shaped building in London, UK. Not sure about how to obtain the equation of the curve but I have the data points that ...
1
vote
2answers
53 views

How we can prove that: $\sum _{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot \log(2)$?

$f:\left[0,1\right]\rightarrow R,\:f(x)=\frac{1}{1+x}$ and we have to show that $\sum_{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot\log(2)$. What I know is just that: $n\cdot \log(2)=\int_0^1 ...
2
votes
1answer
32 views

An integral related to the derivative of Legendre polynomials

I want to calculate the integral $$ I=\int_{-1}^{1} \Big(\frac{\mathrm{d}P_{n+1}(t)}{\mathrm{d}t}\Big) \Big(\frac{\mathrm{d}P_{m+1}(t)}{\mathrm{d}t}\Big) \mathrm{d}t $$ where $P_n(t)$ is Legendre ...
1
vote
1answer
65 views

Definite integral involving 2015

Evaluate $$\displaystyle\int_{2}^{2014} \frac{\log \left( 2015 - x\right )}{\log \left( 2015 - x\right ) + \log \left( x - 1\right )} \mathrm{d}x$$ I got the solution using software, and it is a ...
0
votes
0answers
49 views

how to solve this integral involving any square root

how to solve the integral $\int\sqrt{\alpha+\beta e^{\gamma t}}dt$ i got this integral from the problem Given that the velocity $v$ of a body $t$ segonds after passing a point $O$ is found by ...