Tagged Questions

Questions about the evaluation of specific definite integrals.

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0
votes
1answer
26 views

Proving a Simple Integral with Exponents

Let $f$ be differentiable in $[a,b]$. How can I show that $$\exp\left(\frac{1}{b-a} \int_a^b f(x)dx \right) \le \left(\frac{1}{b-a}\right) \int_a^b \exp(f(x)) dx $$
0
votes
1answer
23 views

Evaluating $\int_0^{\pi/3}\cosh^2\left(x/\sqrt{2}\right)\tan^3x \:dx$

I've been told that this integral admits a closed form $$ \int_0^{\Large\pi/3}\cosh^2\left(x/\sqrt{2}\right)\tan^3x \:dx$$ But an integration by parts with $u'(x)=\cosh^2\left(x/\sqrt{2}\right)$ and ...
2
votes
0answers
40 views

How to evaluate $\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$

Evaluate $$\int_0^1 (\arctan x)^2 \ln(\frac{1+x^2}{2x^2}) dx$$ I substituted $x=\tan\theta$ and got $\int_\frac{\pi}{4}^0 (\theta)^2\frac{\ln(2\sin^2\theta)}{\cos^2\theta} d\theta$ ...
-3
votes
2answers
55 views

Find this integral $\int_{0}^{1}f(x)dx$ [on hold]

let the function $$f(x)=\begin{cases} 1&x\in\{1,\dfrac{1}{2},\dfrac{1}{3}\cdots,\dfrac{1}{n},\cdots\}\\ 2&x\in other \end{cases}$$ Find this integral $$\int_{0}^{1}f(x)dx$$ where this ...
5
votes
0answers
80 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
0
votes
1answer
28 views

Table of Integrals. Which one to use?

I just finished my exam and there was this question. It asks us for which form in the table of integrals to use. $\int(x-3)\sqrt(6+6x-x^2)$ I did completing the square and got into this form ...
1
vote
1answer
30 views

$\int_{-\pi}^{\pi}\sinh(x) \sin(nx)dx$, using complex transformation.

$$\int_{-\pi}^{\pi}\sinh(x) \sin(nx)dx$$ This integral can be done by using integrate by parts twice. However, I was thinking if this can be done by using some complex transformation (I haven't done ...
2
votes
1answer
42 views

using contour integrals

Let $ \gamma (t)= e^{it} $ where $0 \leq t \leq 2 \pi.$ Evaluate $\int_{\gamma}$ $e^{z}$ $dz$ . Use the result to show that $\int_{0}^{2\pi} e^{\cos(t)}\cos(t+ \sin(t)) dt = 0$. I have worked out ...
2
votes
0answers
25 views

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
0
votes
1answer
23 views

Strict upper and lower bounds of a sum (Big-Theta)

I am trying to find a function f(k) such that $S_k=\sum_{n=1}^{k^2-1}(\lfloor\sqrt{n}\rfloor)=\Theta(f(k))$. What I have done so far: Ignoring the floor asymptotically we get: ...
5
votes
2answers
46 views

Find $\mathcal{L}\left\{\cos^3\left(t\right)\right\}$

I began by breaking the problem up as follows: \begin{align} \mathcal{L}\left\{\cos^3\left(t\right)\right\}=\int_0^\infty e^{-st}\cos^3\left(t\right)\:dt & = \int_0^\infty ...
0
votes
3answers
39 views

Proof with Fundamental Theorem of Calculus

If $f'(t)\leq 10$ for $0 \leq t \leq 5$ and $f(0)=3$: How I can explain with $\int_a^b f'(t)\,dt=f(b)-f(a)$ what the maximum of $f(5)$ is?
0
votes
2answers
29 views

Proof with integral properties

I'm trying to explain/looking for an answer whether a positive function $u=f(t)$ exists, for which $\int_{t=0}^{1}u\,dt = \int_{t=1}^{0}u\,dt$ is true. As we all know, the correct theorem is ...
-2
votes
2answers
54 views

Find $\int_0^{2\sqrt{\pi}}\int_{x/2}^{\sqrt{\pi}}\sin(y^2)dydx$

Find $$\int_0^{2\sqrt{\pi}}\int_{x/2}^{\sqrt{\pi}}\sin(y^2)dydx$$ Not sure how to start
15
votes
6answers
222 views

Evaluate $\int_0^1 \frac{x^k-1}{\ln x}dx $ using high school techniques

Is there a way to compute this integral, $$\int_0^1 \frac{x^k-1}{\ln x}dx =\ln({k+1})$$ without using the derivation under the integral sign nor transforming it to a double integral and then ...
0
votes
1answer
31 views

Integral from $e^{-itx}$ over $\mathbb R$

As a part of my task considering characteristic functions I have to compute $\int_{\mathbb R} e^{-itx}dx$ The result i get is $\frac{1}{-it}e^{-itx}|^{\infty}_{-\infty}$, but I don't really know ...
7
votes
4answers
125 views

Evaluate $\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx$ (solution verification)

I tried to find the integral $$I=\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx \tag1$$ I substituted $x=t^2, 2tdt=dx$ and chose $\log x$ and $\sqrt{x}$ to be principal values. We have ...
2
votes
2answers
68 views

Closed form for $\int_0^\infty e^{-x}\sin^a(x)dx$

Can we find a closed form for $$I(a)=\int_0^\infty e^{-x}\sin^a(x)dx$$ Mathematica can easily find closed form for integer $a$: \begin{align*} I(0)&=1\\ I(1)&=1/2\\ I(2)&=2/5\\ ...
3
votes
1answer
42 views

What am I doing with this triple integral?

I am new here and hope my question is clear and is straight to the point. The following is a form of an integral I am trying to compute. $$\int_{x}\int_{y}\int_{z} f(x,y,z) g(x,y)\ dz \ dy \ dx \ ...
3
votes
0answers
18 views

Integral of Bessel function with Gaussian over a quadratic

I need help with the following integral: $$ \int_{0}^{\infty} \frac{J_0(ax)xe^{-bx^2}}{1-cx^2}dx $$ Where $ J_0(x) $ is a Bessel function of the first kind (of zero order). I've looked up a few ...
1
vote
0answers
14 views

$ \int_0^\infty dx\, x^{-3/4} \ln(1+x) \operatorname{Li}_2 \left( -\frac 1 x \right)$

How to show the following identity from Wolfram? $$ \int_0^\infty dx\, x^{-3/4} \ln(1+x) \operatorname{Li}_2 \left( -\frac 1 x \right) =-2\pi \sqrt 2\left[\frac{5 \pi^2}{3} + 16 \left(3 \ln 2 + G - 4 ...
8
votes
2answers
76 views

Closed- form of $\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$

Is there a possibility to find a closed-form for $$\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$$
1
vote
0answers
25 views

Surface area with double integral - how to parameterize?

Problem: Find the surface area of the part of the cylinder $x^2+z^2 = a^2$ that is inside the cylinder $x^2+y^2 = 2ay\;$ and also in the positive octant $( x \ge 0, y\ge 0, z\ge 0$). Assume $a > ...
0
votes
0answers
16 views

The Area of Right Triangle and Integral

Consider that we are dealing with a right triangle with constant base ($B=B_1$ and $\frac{dB}{dt}=0$). The values (of $B=B_1$ and of $\frac{dB}{dt}=0$) are maintained by nature to be constant from ...
-1
votes
0answers
23 views

The Area of Right Triangles and Fundamental Theorem of Calculus

Consider that we are given a right triangle with constant base $B=B_1$ and $\frac{dB}{dt}=0$. The values (of $B=B_1$ and of $\frac{dB}{dt}=0$) are maintained by nature to be constant from the ...
3
votes
0answers
111 views
+50

A Challenge on One Integral Problem

I want to show that if $$I(t)=\int_0^\infty e^{-t^2x}\,\frac{\sinh(2tx)}{\sinh(x)}\,dx$$ then for $t^2\neq1$ $$I(t)=4t\sum_{n=0}^\infty \frac{1}{\left(2n+1+t^2\right)^2-4t^2}$$ Finally, show that ...
1
vote
1answer
13 views

Volume of a split log

When solving the following problem, I could not understand why my reasoning came up with an answer that's different than the one on the solution's manual. Question: Consider $(x,y,z)$ such that ...
5
votes
2answers
94 views

how to integrate $\;I(a) = \int_0^1 \frac{\ln(1-a^2x^2)}{ x^2\sqrt{1-x^2}}dx$

How to solve this integral? $$I(a) = \int_0^1 \frac{\ln(1-a^2x^2)}{x^2\sqrt{1-x^2}}\,dx$$
2
votes
1answer
68 views

Integral $\int_0^{2\pi}\frac{dx}{2+\cos{x}}$

How do I integrate this? $$\int_0^{2\pi}\frac{dx}{2+\cos{x}}, x\in\mathbb{R}$$ I know the substitution method from real analysis, $t=\tan{\frac{x}{2}}$, but since this problem is in a set of ...
2
votes
0answers
26 views

Resummation of a series from an integral

Let's consider the integral $$\int_0^{\infty}e^{-gx^2-x}dx$$ If I'm not mistaken, $$\int_0^{\infty}e^{-gx^2-x}dx=\frac{\sqrt \pi}{2\sqrt g}e^{1/4g}\left(1-\mathrm{erf}\left(\frac{1}{2\sqrt g}\right) ...
2
votes
1answer
91 views

How to integrate $\int_0^1 \sqrt{-x^6+x^4-x^2+1}\:dx$

How do I integrate \begin{equation} \int_0^1 \sqrt{-x^6+x^4-x^2+1}\:dx, \end{equation} which has arisen from a problem I'm working on? I've noticed I can do the following: \begin{align} ...
3
votes
2answers
146 views

Evaluation of $\int_0^1 \frac{x^2}{2(2-x^2)(1+x^2) + 3\sqrt{(2-x^2)(1+x^2)}}\,dx$

How does one evaluate the following integral? $$\int_0^1 \frac{x^2}{2(2-x^2)(1+x^2) + 3\sqrt{(2-x^2)(1+x^2)}}\,dx$$ This is a homework problem and I have been evaluating this integral for hours yet ...
1
vote
2answers
40 views

Evaluate $\int_0^1\int_y^\sqrt{y}\frac{y}{\sqrt{x^2+y^2}}\,dx\,dy$

Not sure how to simplify this and start working it out. Any help and step through would be much appreciated thanks! Ive gotten to ...
0
votes
0answers
50 views

Prove $\int_0^{1/2}\frac{\log(x)\log(1-x)\log(1-2x)}{x(1-x)}dx=\frac{-\pi^4}{96}$ [on hold]

How can I prove the following identity? $$\int_0^{1/2}\frac{\log(x)\log(1-x)\log(1-2x)}{x(1-x)}dx=\frac{-\pi^4}{96}$$
3
votes
1answer
56 views

Proof $f(x)\equiv 0$

Let $f\in C ((-\infty,+\infty)). $ If $ \forall a,b\in(-\infty,+\infty),\int_{a}^{b}f^{2}(x)dx \leq f(a)+f(b), $ then$f(x)\equiv 0$. I can prove $f(x)\geq 0,$ so I want use the reduction to ...
3
votes
2answers
65 views

Find integral $\int^1_0 \sqrt{1-x^2}\,\mathrm dx $ without antiderivative

I don't think integrals are terrible when it comes to finding the antiderivative. But when I just can't do that, I get stuck. Can anyone show me how they'd solve this? $$\int^1_0 \sqrt{1-x^2} dx $$
0
votes
0answers
13 views

Evatuating integral $\int_{-\infty}^{+\infty} 2\chi^2(\rho+v(t - \tau))(V_1 + V_2 \sin(\omega t))e^{-\chi^2(\rho+v(t - \tau))^2}dt$

I am taking analytical mechanics course, I need to solve integral $$\int_{-\infty}^{+\infty} 2\chi^2(\rho+v(t - \tau))(V_1 + V_2 \sin(\omega t))e^{-\chi^2(\rho+v(t - \tau))^2}dt$$ The textbook says ...
0
votes
0answers
19 views

Evaluating integral $\int_{-\infty}^{+\infty} 2\chi^2(\rho+v(t - \tau))(V_1 + V_2 \sin(\omega t))e^{-\chi^2(\rho+v(t - \tau))^2}dt$

I am taking analytical mechanics course, I need to solve integral $$\int_{-\infty}^{+\infty} 2\chi^2(\rho+v(t - \tau))(V_1 + V_2 \sin(\omega t))e^{-\chi^2(\rho+v(t - \tau))^2}dt$$ The textbook says ...
1
vote
0answers
22 views

Integrating a linear function with values from another function

Lets say I have a function like this and call it $f(\lambda)$: And I have another function, that is linear: $y_1=K*a$ where $a$ in this function is known and it is a constant. Now, how would I ...
4
votes
1answer
43 views

The Cosine and an Enigmatic Parabola

In the interval $[-\frac{\pi}{2},\frac{\pi}{2}]$, the cosine function superficially resembles an inverted parabola of the form $-ax^2+1$: I wanted to know more and computed the $L^2$ norm ...
7
votes
3answers
93 views

Finding $\int_0^{\pi/4}\sqrt{1+\left( \tan x\right)^2}dx$

I would like to understand all the steps to find out this integral $$ \int_0^{\pi/4} \sqrt{1+\left( \tan x\right)^2} dx$$ Wolfram Alpha returns: $$ \frac12 \log(3+2 \sqrt2) = 0.881373587019543...$$ ...
8
votes
5answers
134 views

Evaluate $\int\limits_{0}^{\infty}\dfrac{1-e^{-t}}{t}e^{-st}\;dt$

This is laplace transform of $\dfrac{1-e^{-t}}{t}$ and the integral exists according to wolfram Do i get any help/hints about how to work this ? I have been trying integration by parts with different ...
0
votes
1answer
27 views

another definite integral

I'm having trouble solving this equation.. $$\int_0^{e-1}\frac{1}{x+1}\,\mathrm dx$$ I have tried the substitution method with $u= x+1$ and $du= 1dx$ $$\int_\ udu$$ then I get ...
-1
votes
3answers
63 views

Evaluating $ \int \frac{3}{x^2+8x+17}dx$ [closed]

How to find this integral: $$\displaystyle \int \dfrac{3}{x^2+8x+17}dx$$
0
votes
1answer
36 views

Integrating an exponential function

How do I integrating this function- I=$\int \exp{\mathrm{\frac{(-1)(z^2-k)^2}{2}}}dz\\$ limit is from -infinity to +infinity I tried a lot by nothing worker!Pls help!
7
votes
4answers
104 views

$\int_{0}^{1}\sqrt{x^2+1}\,dx$

The integral $\displaystyle \int_{0}^{1}\sqrt{x^2+1}\,dx$ can be evaluated with the standard technic of sub $u=\tan \theta$. However, in the book says evaluate the integral without trigonometric ...
0
votes
2answers
44 views

Double integral: Stokes' theorem

Honestly, I have no idea what Stokes' theorem is. I only know that the circulation can be found by this theorem. Would anyone mind helping me?
3
votes
2answers
53 views

An Inequality involving integration

Show that $$\int_{0}^{\pi} \left|\frac{\sin nx}{x}\right| dx \ge \frac{2}{\pi}\left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right)$$ How do I go about proving this inequality ?
4
votes
2answers
90 views

How to solve $\int_{0}^{2\pi} \frac{\cos(50x)}{5+4\cos(x)} dx\,?$

I encountered this integral and tried to solve it. As you can expect I could not solve this and thought I will ask it here. The integral is: $$\int_{0}^{2\pi} \frac{\cos(50x)}{5+4\cos(x)}\, dx$$ I ...
1
vote
0answers
10 views

Integral of product of Hermite functions with rescaled weights.

Let $$h_{k}(x)=c_{k}(-1)^k e^{\frac{x^2}{r^2}}\frac{d^k}{dx^k}e^{-\frac{x^2}{r^2}}$$ be the standard Hermite polynomials, rescaled with a given parameter $r>0$. The normalizing constant ...