Questions about the evaluation of specific definite integrals.

learn more… | top users | synonyms (1)

0
votes
1answer
27 views

$\int_{0}^{2\pi}\int_{0}^{\pi}\sin^3y \ e^{3\cos x\sin y+4\sin x\sin y}\,dy\,dx$

I am working on this double integral $\displaystyle\int_{0}^{2\pi}\int_{0}^{\pi}\sin^3y \ e^{3\cos x\sin y+4\sin x\sin y}\,dy\,dx$ so far, I don't know how to start. Can someone give a hint? Thanks ...
3
votes
2answers
45 views

How to use Stokes Theorem to evaluate $\int_{S} \text{curl} F\cdot d\mathbf{S}$

Let F = $( yz, 0, x)$ and $S$ is the portion of the plane ${x\over2} + {y\over3} + z = 1$ where $x, y, z \ge 0$, oriented with an upward pointing normal then prove: $$\int_{S} \text{curl} F\cdot ...
1
vote
2answers
32 views

General Solution for the Gravity Between Two 3D Triangles

I would like to find the general solution for the gravity between two (flat) triangles in 3D, including the location $(x,y,z)$ where this force should be applied (in order to later account for ...
1
vote
0answers
43 views

Inequality involving integral of $\Gamma(x)$

The graph of $\frac{e^x}{\Gamma(x+1)}$ is somewhat bell-shaped. I think the proof of the following requires an understanding of the integral of this function that I can't glean from either Mathematica ...
7
votes
6answers
172 views

Integration by differentiating under the integral sign $I = \int_0^1 \frac{\arctan x}{x+1} dx$

$$I = \int_0^1 \frac{\arctan x}{x+1} dx$$ I spend a lot of my time trying to solve this integral by differentiating under the integral sign, but I couldn't get something useful. I already tried: ...
0
votes
0answers
29 views

Definite integration $\int_{0}^\infty \frac{{x^2}}{e^{\beta {\big(\sqrt{x^2 + m^2}}- \nu\big)} + 1} dx$

$$I = \int_{0}^\infty \frac{{x^2}}{e^{\beta {\big(\sqrt{x^2 + m^2}}- \nu\big)} + 1} dx = ?$$ If a constant is added to the exponential in the denominator along with a square root in the exponent and ...
4
votes
3answers
210 views

Integration $\int_{m}^\infty {x}{\sqrt{x^2 - m^2}}e^{-\beta x} dx$

How can the following integration be performed? Does it involve Bessel functions?$$\int_{m}^\infty {x}{\sqrt{x^2 - m^2}}e^{-\beta x} dx$$ EDIT: Actually, the original question is: $$\int_{0}^\infty ...
2
votes
2answers
83 views

How to evaluate the integral $\int_1^{\infty}[u^\alpha-(u-1)^{\alpha}]^2du$?

The integral \begin{equation} I=\int_1^{\infty}[u^\alpha-(u-1)^{\alpha}]^2du,\quad -1/2<\alpha<1/2 \end{equation} comes from the stochastic process fractional Brownian motion. Can someone show ...
3
votes
2answers
86 views

Proof of a closed form of $\int_0^1(-\ln x)^ndx$

$$\int_0^1(-\ln x)^ndx$$ Is there a step-by-step solution to a closed form of this expression? I've tried using different representations to re-write the expression but I couldn't find anything I knew ...
0
votes
2answers
78 views

Resolution of limits like this: $\lim_{n\to+\infty}\int_{0}^{\frac{\pi}{2}} e^{-n \sin(x)} \,dx $ [on hold]

Good morning. Can you give an help to solve these limits? I have thought of using the uniform convergence $$a)\lim_{n\to+\infty}\int_{0}^{\frac{\pi}{2}} e^{-n \sin(x)} \,dx \\ ...
4
votes
3answers
54 views

$\int_{-\pi/2}^{\pi/2} dx \, \sin^{2n} x $

While reading a physics paper I came cross the following set of integrals. $$\int_{-\pi/2}^{\pi/2} dx \, \sin^{2n} x $$ I tried using De Moivre identity but not sure about the conclusion: $$ ...
5
votes
3answers
119 views

How to integrate $f(x) = \frac{1}{a + b \cos x + c \sin x }$ over $x \in (0,\pi/2)$

Conjecture 1 $$ \begin{align*} I_{T}=\int_0^{2\pi} \frac{\mathrm{d}x}{a + b \cos x + c \sin x} & = \frac{2\pi}{\rho} \tag{1} \\ I_{T/4} = \int_0^{\pi/2} \frac{\mathrm{d}x}{a + b \cos x + ...
0
votes
2answers
40 views

Inequality in Integral

Show that $\dfrac{28}{81}<\int_0^\frac{1}{3}e^{x^2}dx<\dfrac{3}{8}$. It would be great if a solution based on the Mean Value Theorem for Integrals is posted.
1
vote
3answers
112 views

Demonstration of $\int_{a}^b f(x) \,dx= 0 \Rightarrow f(x)\equiv0 $ [on hold]

Good morning, Can you give me a help to demonstrate this proposition: $f$ is a continuous and not negative function on the interval $[a,b] \ a,b \in \Re $, Demonstrate: $$\int_{a}^b f(x) \,dx= 0 ...
2
votes
1answer
62 views

How to compute the integral $ \int_0^\pi \lfloor\cot (x)\rfloor dx $

I have been given to compute $$ \int_0^\pi \lfloor\cot (x)\rfloor dx$$ Where, $ \lfloor$ $\rfloor$ is floor function. Now since floor function is discontinuous we need to break out Integral in ...
5
votes
1answer
70 views

Evaluate $\int_{-2}^{-1}\frac{\text{d}x}{\sqrt{-x^2-6x}}$.

Problem statement [from Charlie Marshak's Math GRE Prep Problems]: Evaluate $\displaystyle \int\limits_{-2}^{-1}\dfrac{\text{d}x}{\sqrt{-x^2-6x}}$. My work: notice that $$\begin{align} -x^2-6x ...
-2
votes
3answers
85 views

Comparison of the integral $\int_{0}^{10000} x^{1/2} \,dx$ and the sum $\sqrt{1} + \sqrt{2}+ \cdots+\sqrt{10000}$ [on hold]

I am stuck in this problem Let $S = \sqrt{1} + \sqrt{2}+ \cdots \sqrt{10000}$ and $$I =\int_{0}^{10000} x^{1/2} \,dx$$ Show that $I \leq S \leq I +100$
0
votes
2answers
42 views

How to decide limits of integral in Riemann's sum

I need to calculate $$\lim_{n \to \infty}\frac{1}{\sqrt{n^2 -0^2}}+\frac{1}{ \sqrt{n^2 -1^2}} + \frac{1}{ \sqrt{n^2 -+2^2}}+.....+ \frac{1}{ \sqrt{n^2 -(n-1)^2} }$$ It can be written as $$ ...
18
votes
4answers
422 views

Integrating $\int_0^\pi \frac{x\cos x}{1+\sin^2 x}dx$ [duplicate]

I am working on $\displaystyle\int_0^\pi \frac{x\cos x}{1+\sin^2 x}\,dx$ First: I use integrating by part then get $$ x\arctan(\sin x)\Big|_0^\pi-\int_0^\pi \arctan(\sin x)\,dx $$ then I have ...
3
votes
2answers
55 views

Demonstration of $\int_{-a}^a \frac{f(x)}{1+e^x} \,dx= \int_0^a f(x) \,dx$ [duplicate]

Good morning, Can you give me a help to demonstrate this proposition: $f$ is an even and continuous function on the interval $[-a,a], a>0$. Demonstrate: $$\int_{-a}^a \frac{f(x)}{1+e^x} \,dx= ...
2
votes
2answers
32 views

Surface Area (Integration)

Find the surface area of the object by rotating $y = 4+3x^2$ about the $y$-axis, where $1 \leq x \leq 2$. I've been using the formula: Surface area: Definite integral of $$2\pi ...
9
votes
3answers
226 views

What are other methods to evaluate $\displaystyle\int_0^1 \sqrt{-\ln x} \ \mathrm dx$

$$\int_0^1 \sqrt{-\ln x} dx$$ I'm looking for alternative methods to what I already know (method I have used below) to evaluate this Integral. $$y=-\ln x$$ $$\bbox[8pt, border:1pt solid ...
0
votes
0answers
13 views

application of line integral with repect to x or y.

how line integrals with respect to x or y are used. I'am looking for how they are useful ? what kind of math or real life problems they solve? \begin{align} \int_C f(x,y)\,dx &:= ...
2
votes
0answers
24 views

Interpretation of a line integral with respect to x or y .

i read about some interpretation ideas in Interpreting Line Integrals with Respect to $x$ or $y$ and i was wondering if the interpretation given below is right or not ? lets say we have : ...
5
votes
3answers
174 views

Integrate $I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$

I have a problem with this integral. It seems that solution has to be simple, but I couldn't find out. $$I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$$ I tried using integration by parts and ...
0
votes
1answer
55 views

Double Integral Problem - Transforming limits

The double integral $\int_0^2\int_x^{4-x} f(x,y) dy dx $ under the transformation $ u = x+2$, $v = y- 2x$ is transformed into? My attempt: I used the equations $y=x$ and $y=4-x$ to get the relations ...
5
votes
3answers
128 views

Is the function $f(x) = \begin{cases} 1 & \text{$x\in\Bbb Q$} \\[2ex] 0 & \text{$x\notin\Bbb Q$} \end{cases}$ Riemann integrable?

$f(x) = \begin{cases} 1 & x\in\Bbb Q \\[2ex] 0 & x\notin\Bbb Q \end{cases}$ Is this function Riemann integrable on $[0,1]$? Since rational and irrational numbers are dense on $[0,1]$, no ...
8
votes
4answers
199 views

Expression for $\int_0^1 x^n(1-x)^{n}/(1+x^2) \ dx$

An answer to this question makes clever use of an integral of this form: $$\int_0^1 \frac{x^n(1-x)^n}{1+x^2} dx$$ Is there a closed form for this for arbitrary positive integer $n$? (I expect this ...
0
votes
3answers
76 views

$\int_{0}^{1} (x-1)\sqrt{1-x} dx$ without Parts etc..

Evaluate: $$\int_{0}^{1} (x-1)\sqrt{1-x} dx$$ Without the use of integration by parts (1) My initial thought is, can we use series for either one of these? Can we find a series (about x=0) for ...
3
votes
2answers
74 views

How do I evaluate the following integral $\int_{-\infty}^{\infty} e^{-\sigma^2 x^2/2}\; \mathrm dx$? [duplicate]

How do I evaluate the following integral $$\int_{-\infty}^{\infty} \exp\left(-\frac{\sigma^2 x^2}{2}\right) \mathrm dx\;?$$ How is it even possible to find an antiderivative? The integral is ...
0
votes
0answers
22 views

Parametrization of a 3D surface

While solving the following problem: The $x$ and $y$ coordinates of a point on the $Paraboloid$ $2z = x^2/a + y^2/b$ are expressed in the form $x = atanθ cosγ $, $y = btanθ sinγ $ where $θ$ is the ...
2
votes
1answer
66 views

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

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
0
votes
1answer
19 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 ...
0
votes
1answer
64 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?
0
votes
0answers
25 views

Find the flux through a closed volume with the divergence theorem and using the definition

Given the vector field F(x,y,z)=(xy,xy,z) and $D= \{(x,y,z) \in R^3 : x^2 + y^2 + z^2 \le 4, x^2 + y^2 \le 1, z\ge 0 \}$ Find the flux through ...
7
votes
2answers
82 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 ...
0
votes
0answers
49 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
votes
1answer
38 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 ...
1
vote
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 ...
-2
votes
1answer
42 views

Evaluating a complex integral using the Cauchy integral formula [closed]

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
votes
2answers
46 views

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

I was asked to find the Laplace transform of the function $\displaystyle f(t)=te^{-t}\sin(2t)$ using only the properties of Laplace transform, meaning, use clever tricks and the table shown at ...
0
votes
2answers
36 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?
5
votes
4answers
197 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 ...
6
votes
0answers
144 views
+50

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

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 ...
7
votes
2answers
125 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
votes
1answer
79 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
votes
1answer
155 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 ...
1
vote
0answers
39 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\\ ...
0
votes
1answer
29 views

Find the area between the two functions--integrals [closed]

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, $y=x^2+3$