Questions about the evaluation of specific definite integrals.

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1answer
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how to calculate the following integral$\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$ [on hold]

calculate the following integral $\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$ I need to very hollowing steps.thank you in advance
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1answer
11 views

Transform integral bounds for multidimensional integral

I need to transform the followin integral borders into something I can use to integrate the argument of $$\int_{R^N, \; \|\vec{x}\| \lt \gamma} f(\|\vec{x}\|) d\vec{x}$$ analytically. The ...
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1answer
55 views

How to check if functions are integrable?

Consider two functions $$ \int_0^1 \frac{1}{e^x-1} dx $$ and $$ \int_0^1 \frac{1}{(e^x-1)^2} dx $$ How to check if these functions are integrable?
7
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2answers
73 views

If f is differentiable on (a,b) continuous at a and f has bounded derivative must f be right differentiable at a?

If $f$ is differentiable on $(a,b)$ continuous at a, and $f$ has bounded derivative, must $f$ be right differentiable at $a$? In case answer to previous question is true, is the statement still ...
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0answers
47 views

I have calculate an integral. Is it correct?

Could you please help me to check this integral? Is it correct? \begin{align} I&=\int \limits_{\gamma_0}^{\infty}\frac{exp(-x D_6)}{(x A_3 +1)(x A_4+1)^{m}}dx \\ &=\int ...
0
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1answer
34 views

Space of continuous functions linear operator eigenvalues

Let $V$ be the vector space of continuous functions from $\mathbb R$ to $\mathbb R$. Let $T$ be the linear operator on $V$ defined as $$(Tf)(x)=\int_0^x f(t)\,dt$$ Prove that $T$ doesn't have ...
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0answers
19 views

Integrating a function by its parameter $\int_{Z_1}^{Z_2}dz = \int_{0}^{x_1}A\cdot y_{(x)}^2dx$

I have the equation $dz = A \cdot y_{(x)}^2dx$ which needs to be integrated. $A$ is a constant $z$ is a function of $y$ $y$ is a function of $x$ I thought of the following (those are the limits ...
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1answer
41 views

Evaluating a complex integral using the Cauchy integral formula [on hold]

I need to evaluate the following integral counterclockwise: $$\oint_{\left | z \right |=\frac{1}{2}} \frac{dz}{(z-1)\sin z} $$ using the Cauchy integral formula
3
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2answers
31 views

Laplace transform of $f(t)=te^{-t}\sin(2t)$

I was asked to find the laplace transform of the function $f(t)=te^{-t}\sin(2t)$ using only the properties of laplace transform, meaning, use clever tricks and the table shown at ...
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2answers
33 views

Evaluation of an integral of some expressions involving fractions

I am stuck in evaluating the following integral: \begin{equation} \int_{0}^{b-a} \frac{1}{\sqrt{u} (a+u)} \,du, \end{equation} where $0<a<b$. Any ideas?
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4answers
168 views

Finding $ \int_0^1 \frac {\ln x}{1+x^2}\mathrm dx $

Today I encountered the problem of how to find $$ \displaystyle\int_{0}^{1} \frac {\ln x}{1 + x^2}\mathrm dx $$ but got no start on it. Is this one of those integrals which we have to approach from ...
4
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0answers
69 views

An infinite series of integrals $\int_{0}^{\eta}\cos nt\log\left(\frac{\cos(t/2)+\sqrt{\cos^2(t/2) -\cos^2(\eta/2)}}{\cos(\eta/2)}\right) dt$

I am reading a paper (sorry, no e-copy) with a number of infinite series, in which each term of the series is an integral of a complicated transcendental function like the one in the title. There ...
6
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2answers
113 views

Closed form of $\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln x}{(1-x)^3}\,\mathrm dx\;\mathrm dy\;\mathrm dz$

While trying to find several references to answer Pranav's problem, I encounter the following multiple integrals $$I=\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln ...
5
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1answer
77 views

Multiple integrals involving product of gamma functions

The following integral was posted a few days back on Integrals and Series forum: $$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos ...
5
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1answer
139 views

How evaluate the following hard integrals?

Prove: $$\displaystyle\int_0^{\frac{\pi}{4}}{\,x}{\,\arctan\sqrt{\frac{\cos2x}{2\sin^2x}}}dx=\frac{\pi}{96}[{\pi^2}-6\ln^22]$$ And ...
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0answers
38 views

Clarifying a step in an integration solution

In the accepted answer here, the first two steps in computing the integral are \begin{align} \mathcal{I} =&\frac{1}{2}\int^\infty_0\ln(1-e^{-2x})\ln\left(\frac{x^2}{\pi^2+x^2}\right)\ {\rm d}x\\ ...
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1answer
29 views

Find the area between the two functions--integrals [on hold]

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region $y=5x^2$ and, $y=x^2+3$
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2answers
34 views

Finding the limit of this integral: $\lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$ if $q<p+1$

I am trying to find the following limit provided: $q<p+1$: $$ \lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$$ Dividing by $n x^q$ so we have $$\dfrac{n x^p+x^q}{x^p+n ...
1
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1answer
50 views

How to compute $\int^{1}_{-1}f(x)dx$?

I need to compute $\displaystyle\int^{1}_{-1}\,{\rm f}\left(\, x\,\right)\,{\rm d}x$, where $$ \,{\rm f}\left(\, x\,\right) =\left\{\begin{array}{lcrcl} x & \mbox{if} & x & \leq & 0 ...
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0answers
170 views

Evaluate $\int_0^{\pi/2} \frac{\ln\left(e^{2x} + 1\right)}{1 + \sin2x}\mathrm dx$

Here's my Xmas gift to all of you! I just encountered a very tough integral. $$\int_{0}^{\pi/2} \frac{\ln\left(e^{2x} + 1\right)}{1 + \sin2x}\mathrm dx$$ I have tried for a few hours. This task is ...
5
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2answers
78 views

Closed form of a Definite Integral [duplicate]

I attempted to integrate the following function from a practice problem in my Calculus textbook: $$\displaystyle \int_{0}^{\frac{\pi}{2}}{\frac{1}{1+\tan^\sqrt{2}(x)}} \ {\rm d}x$$ I failed to find ...
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1answer
80 views

Evaluating $\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$

How to calculate this integral? $$\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$$ I suppose that it should be parted like this: $$\int_0^{1} \frac{dx}{\sqrt[3]{2x^2-x^3}} + \int_1^{2} ...
11
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4answers
192 views

Proving $\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} dx=1-\frac{\gamma}{2}-\ln2$

Nowadays I encounter an integral which is difficult for me to evaluate it. Please help me to evaluate it. Thank you. $$\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} ...
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0answers
29 views

How to calculate the volume of a tetrahedron?

Suppose that $$ I=\iiint_{V}f(x,y,z)dxdydz $$ where $f(x,y,z)$ is a continuous function, $V$ is a tetrahedron whose vertices are $P(2,2,0), A(-2,0,0), B(0,0,2)$ and $C(1,1,3)$. I want to ask you how ...
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1answer
19 views

Area of the region inside $r = 1 - \cos(\theta)$ and also inside $r = \cos(\theta)$

Pretty simple polar integration question that I've been having trouble with... The question says it all. I identified the limits of integration by setting $1 - \cos(\theta) = \cos(\theta)$ so that ...
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0answers
63 views

Calculating an integral with sine, cosine

I've recently calculated the Fourier transform of $\dfrac{\sin \pi ax}{\pi x}$. Now I'm trying to calculate $$\int _{\mathbb{R}} \frac{\sin ^2 \pi ax}{\pi ^2 x^3} \cos \pi bx\;\mathrm dx$$ The ...
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0answers
27 views

How to compute $\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$

For a project I want to get a closed form solution of $$\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$$ Here $p \in \mathbb{N},\; d\ge3, \; d\in\mathbb{N}$ and $P_n^d$ is the associated ...
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1answer
54 views

How to calculate the area between $y=e^{-x}$, $y=x$ and $x=0$

My problem is that little point that comes from the equation $$e^{-x} = x$$ I can't solve that one. Is there another way without knowing that point or a way to calculate it? Thanks in advance!
4
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2answers
61 views

Easier way to solve $\int_0^1 \frac{dx}{\lfloor{}1-\log_2(x)\rfloor}$

This problem showed up in the MIT integration bee last year: $$\int_0^1 \frac{dx}{\lfloor{}1-\log_2(x)\rfloor}$$ Basically, after doing a lot of tedious work I graphed out part of the function and ...
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0answers
29 views

Stokes' theorem to evaluate an integral

Let $C$ be the intersection curve of the parabolic sheet $y=x^2$ with the cylinder $x^2+z^2=4$, oriented clockwise when viewed from the positive $y$-axis. Apply Stoke's Theorem to the integral $$ ...
5
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3answers
107 views

Putnam definite integral evaluation $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$

Evaluate $$\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$$ Source : Putnam By the property $\displaystyle \int_0^af(x)\,dx=\int_0^af(a-x)\,dx$: $$=\int_0^{\pi/2}\frac{(\pi/2-x)\sin ...
5
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2answers
86 views

Prove that $\int_0^\infty\frac{x^n}{1+e^{x-t}}\mathrm{d}x = \frac{t^{n+1}}{n+1} + o(t^n)$, when $t \to \infty,\,n\in\Bbb{R}^+$

I hae to prove that $$\int_0^\infty\frac{x^n}{1+e^{x-t}}\mathrm{d}x = \frac{t^{n+1}}{n+1} + o(t^n), \quad\text{ when } t \to \infty,\,n\in\Bbb{R}^+$$ where $o(\cdot)$ is the Little-o notation. What ...
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2answers
37 views

Finding the mass of a cone using triple integral

I have a density $\rho(x,y,z) = 3-z$ and have converted my given information to form a triple integral equation for finding the volume of my cone in cylindrical coordinates and have found the volume ...
0
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1answer
36 views

Volume bounded by two solids

Can somebody help me get started in the right direction for this question involving volume? The question is "Find the volume of the solid region inside the hemisphere $x^2 + y^2 + z^2 =6, z<0$ but ...
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1answer
45 views

Evaluate $\int_0^1\int_x^1 e^{x/y} dy\,dx$

I need some help to solve the following: $$\int_0^1\int_x^1 e^{x/y} dy\,dx$$ I guess it is related with change of variable, but I can't figure out which one. Thanks in advance. Regards.
2
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1answer
43 views

Double integral with two parameters $\int_{x_1=1}^{x_2=2}\int_{y_1=0}^{y_2=x}\arctan\left(\frac{y}{x}\right)\,dx\,dy$

Given the following integral: $$\int_{y_1=0}^{y_2=x}\,dy\int_{x_1=1}^{x_2=2}\arctan\left(\frac{y}{x}\right)\,dx$$ I thought of using $u$-substitution: $$\begin{align} u &= \frac{y}{x} \\ w ...
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1answer
21 views

Volume of a solid formed as vertical limit goes to infinity

Here's the question: The way I have it set up currently is as follows: $V = \pi \lim_{a \to \infty} \int_1^a (a-1)^2 - (\frac{1}{\sqrt{x^5}} - 1)^2$ But how do I go from here? And is the working ...
1
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2answers
66 views

Intuitive but hard question about an integral?

Let $f \colon [0,1]\rightarrow \mathbb{R}$ be a function with continuous derivative such that $f(1)=1$. Evaluate $$\lim_{y\rightarrow \infty}\int_0^1yx^yf(x)dx$$
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0answers
29 views

Any way to simplify integral of Confluent Hypergeometric Function of the First Kind?

The integral is this: $$\int_{-\log n}^{0}e^{t(1-s)} \cdot z \cdot {}_1F_1(1-z, 2, t) dt $$ Is there a way to write this in terms of special functions that eliminates the integral and doesn't use ...
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0answers
15 views

Is it always possible to use Chasles to decompose an integral ?

Well, I'm in "classe préparatoire" and I always learn that if f is a continuous fonction integrable on [a,b] and if c is in [a,b] then with Chasles relation we have : $$ \int_a^b f(x) dx = \int_a^c ...
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0answers
47 views

How do we calculate naturally the following integral :

Do you know a natural method to calculate the following integrals: $$ I = \int_{\mathbb{P}^{1} (\mathbb{R})} \dfrac{1}{x} dx\quad\text{and}\quad J = \int_{\mathbb{P}^{1} (\mathbb{C})} \dfrac{1}{z} dz. ...
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1answer
36 views

Is it possible to find the closed-form expression for $\int_{\alpha}^{\infty} \frac{e^{-At}}{\left(1+ Bt\right)t^m}dt$?

Is it possible find the closed-form expression or represent it in other function for this integral: \begin{align} I= \int \limits_{\alpha}^{\infty} \frac{e^{-At}}{\left(1+ Bt\right)t^m}dt ...
3
votes
1answer
43 views

Handling integrals of trig functions

I'm not sure how to handle the following class of integrals: $I=\int_0^{2\pi}f(\cos(\theta))d\theta$ If I make the change of variables $x=\cos(\theta)$ the new limits of the integral are the same, ...
1
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2answers
76 views

Solving $ \int_{0}^{2\pi} e^{-x} \lvert \sin x\rvert,dx $

Since it involves an absolute value, I assume I need to split it into two cases? For $ 0 \le x \le \pi $ $$ \int_{0}^{\pi} e^{-x} \sin x\,dx $$ and for $ \pi \le x \le 2\pi $ $$ \int_{\pi}^{2\pi} ...
2
votes
1answer
36 views

How to evaluate a double integral with two Dirac functions?

Here I have a problem, is the solution the same if I integrate every one? part by part? $$\int_0^Te^{-(s+\mu\lambda^2 ) t} \int_0^l\left[\delta(x-R)\delta(t-tj)\varphi(x) \, dx\, dt\right]$$ I've ...
2
votes
2answers
48 views

Taylor Expansion of $x\sqrt{x}$ at x=9

How can I go about solving the Taylor expansion of $x\sqrt{x}$ at x=9? I solved the derivative down to the 5th derivative and then tried subbing in the 9 value for a using this equation ...
17
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5answers
266 views

Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$

Today I discussed the following integral in the chat room $$\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$$ where $0\leq a, b\leq \pi$ and ...
1
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1answer
33 views

Riemann integrability of a square of a continuous function

Let, $f(x)$ be continuous in $[0,1]$ such that, $\int_{0}^{1}x^{n}f(x)dx=0$ for $n=0,1,2,3,...$. Then prove that, $\int_{0}^{1}f^{2}(x)dx=0$. First we apply $1^{st}$ M.V.T. of integral calculus & ...
4
votes
1answer
63 views

Reduction formula of primitive $\big(1-\sin^3{x}\big)^n\cos{x}$

I am trying to obtain a reduction formula for $$\int_0^{\pi/2}\big(1-\sin^3{x}\big)^n\cos{x}\;\mathrm dx $$ where $n \in \mathbb{N}$. My attempt is as follows $$\text{let } v = \sin{x}\; \implies ...
5
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0answers
64 views

Find a function that maximizes $\int_{0}^{1}f(x)\,\rm dx$ with given constraints

Find a function $f(x)$ that maximizes the following integral $$\max\int_{0}^{1}f(x)\,\rm dx\quad \text{s.t.}\quad \frac{d}{dx}ln(f(x))<0$$ $f(x)$ also continues, $f:[0,1]\rightarrow R$ and we ...