Questions about the evaluation of specific definite integrals.

learn more… | top users | synonyms (1)

0
votes
3answers
42 views

Prove $\int_0^{2\pi}\frac{3a\sin^2\theta}{(1-a\cos \theta)^4}$ or $\int_0^{2\pi}\frac{\cos\theta}{(1-a\cos\theta)^3}=\frac{3a\pi}{(1-a^2)^{5/2}}$

While doing some mathematical modelling of planetary orbits I have come up with two definite integrals $D_1$ and $D_2$ which appear to produce the same result R when tested with various values of $a$ ...
3
votes
0answers
73 views

To determine a definite integral

I have been trying to solve the following integral $$\int_{0}^{\frac {\pi}{2}} \ln\left (\frac {\ln^2 (\sin x)}{\pi^2+\ln^2 (\sin x)}\right) \frac {\ln \cos x}{\tan x} dx$$ I tried substituting for ...
0
votes
1answer
23 views

Change of limits in definite integral - non constant limit

I have the following definite integral: $\displaystyle \int^{g(a)}_{a} f(x) \, dx$ and I am asked to perform a shift of the variable x, so that it transforms in $x + T$ (T is just some constant). ...
4
votes
1answer
52 views
0
votes
0answers
24 views

volumes and integrals help [on hold]

Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves $y=x^2$ and $y=x^4$ and $y=16$ and $x=0$ about the $y$-axis.
1
vote
1answer
36 views

Proving that $F(x)$ is a constant

This was on a test and i know i was supposed to use 2nd ftoc to prove that $F(x)$ was a constant when $x>0$ $$ F(x) = \int_{0}^{x} \frac{1}{t^2 +1} dt + \int_{0}^{\frac{1}{x}} \frac{1}{t^2 +1} ...
7
votes
2answers
84 views

Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2$

This question inspired me to ask the following. Prove that $$I_n = \int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2,$$ for $\Re(n)>1$. For some ...
3
votes
1answer
81 views

Closed form of $I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx$

Does the integral below have a closed-form: $$I=\int_{0}^{\pi/2} \tan^{-1} \bigg( \frac{\cos(x)}{\sin(x) - 1 - \sqrt{2}} \bigg) \tan(x)\;dx,$$ where $\tan^{-1} (\cdot)$ is inverse tangent function. ...
1
vote
2answers
46 views

What is $\int_0^{\infty} x^2e^{\frac{(x-\mu)^2}{2 a^2}} dx$?

How can we express the integral $\int_0^{\infty} x^2e^{-\frac{(x-\mu)^2}{2 a^2}} dx$ for example by means of the error function? The problem is of course, that the expectation value is shifted and we ...
1
vote
0answers
24 views

Can I do the following when solving my integration??

I appreciate any feedback for my question. I have an integration as follows $$\int_{-\pi}^{\pi}\frac{1}{2\pi} \prod_i \frac{1}{1+ x_ig(\theta)} d\theta $$ I have that $g(\theta)$ is the defined as ...
12
votes
5answers
164 views

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...
1
vote
0answers
19 views

Asymptotic limit of a definite integral

I would like to solve the following integral: $$ I_0 (a,b)= \int_0^1 dx\int_0^{1-x} dz \frac{1}{a z (z-1)+a x z + x(1-b)}$$ in the limit where $b$ is small and $a$ is large ($a$ and $b$ are positive ...
0
votes
0answers
38 views

A question about $f(x)\equiv C$

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
2
votes
2answers
95 views

How to show that $\int_0^1 dx \frac{1+x^a}{(1+x)^{a+2}} = \frac{1}{a+1}$?

From numerical evidence it appears that whenever the integral converges, $$J_a :=\int_0^1 dx \frac{1+x^a}{(1+x)^{a+2}} = \frac{1}{a+1}.$$ For $a \in \mathbb{N}$, I was able to prove this using ...
0
votes
1answer
19 views

Solving the integral which shows the second moment of subtracting two Beta-distributed Random Variables

Peace be upon you In my project I needed to find the second moment of the subtraction of two Beta-distributed random variables. I have computed it and reached to the following integral which I should ...
0
votes
2answers
30 views

Integrating the gamma function

I assumed that $$\Gamma\left(k+\frac{1}{2}\right)=2\int^\infty_0 e^{-x^2}x^{2k}\,dx=\frac{\sqrt{\pi}(2k)!}{4^k k!} \,,\space k>-\frac{1}{2}$$ and that ...
3
votes
1answer
45 views

A suitable integration path for $\cos z/(2 + \cos z)$

I was solving the exercises and got stuck when trying to solve this with tools of residual calculus $$ \int_{0}^{2 \pi} \frac{\cos (z)}{2 + \cos (z)} \, dz = \int_{0}^{2 \pi} f(z) \, dz. $$ Isolated ...
6
votes
3answers
87 views

Integration $I_n=\int_{0}^{1}\frac{dx}{(x^n+1)(\sqrt[n]{x^n+1})}$

$$I_n=\int_{0}^{1}\frac{dx}{(x^n+1)\large\sqrt[n]{\normalsize x^n+1}}$$ Could someone help me through this problem?
2
votes
1answer
60 views

Closed-form of $\int_0^1 \int_0^1 \int_0^1 x^{(y^z)} \,dz\,dy\,dx$

We know that $$\int_0^1 \int_0^1 x^y\,dy\,dx = \ln 2.$$ Do we know a closed-form of $$\int_0^1 \int_0^1 \int_0^1 x^{(y^z)} \,dz\,dy\,dx\,?$$ As a start we know that $$\int_0^1 x^{(y^z)}\,dz = ...
0
votes
0answers
15 views

Setting up a volume-finding calculation

I'm asked to find the volume inside the sphere $x^2+y^2+z^2=25$ and outside the cylinder $x^2+y^2=1$. I approached the volume $V$ in the following way: ...
5
votes
2answers
103 views

Prove using contour integration that $\int_0^\infty \frac{\log x}{x^3-1}\operatorname d\!x=\frac{4\pi^2}{27}$

Prove using contour integration that $\displaystyle \int_0^\infty \frac{\log x}{x^3-1}\operatorname d\!x=\frac{4\pi^2}{27}$ I am at a loss at how to start this problem and which contour to pick. I ...
2
votes
2answers
132 views

How to evaluate $\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x$?

How to evaluate the following integral $$\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x $$ The numerical result is $= -1.38104$ and when I look at it, I have no idea how to ...
7
votes
1answer
69 views

Integral involving hyperfactorial

I'm trying to prove that: $$ \int_0^1 \ln\left(K(x)\right)\space dx =-\zeta'(-1)=\ln(A)-\frac{1}{12} $$ Where $A$ is Glaisher Kinkelin's constant and $K(x)$ is a generalization of the hyperfactorial ...
0
votes
0answers
18 views

Triple integral containing definite integral and exponentials with trigonometric functions

I am attempting to solve the following integral analytically: $$ \int_{z=5i}^{z=1} \int_{t=\csc^{-12}(z)}^{t=2} \int_{\theta=\sin^{t}(z)}^{\theta=t^2} {[\mathrm{e}^{t\cos(\mathrm{e}^{i \theta})} + ...
0
votes
2answers
38 views

Integral with parameter: $\int_{0}^{a}x^2\sqrt{a^2-x^2}dx$

I have the following integral : $$\int_{0}^{a}x^2\sqrt{a^2-x^2}dx$$ I tried to manipulate the integral and then use substitution to get a rational form to arrive at: $$-8a^4\int_0^a ...
0
votes
0answers
29 views

Proving $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)} dx < \frac{1}{e}$

While i was playing around with very weird functions and came across this: $$ \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)}dx \approx 0.3676932086...\approx \frac{1}{e} - ...
1
vote
0answers
26 views

Definite integral substitution question.

Let's say you have $$\int^\infty_0 f(x) dx$$ and you substitute $x=u^2$ so that the integral becomes, $$\int^\infty_0 f(u) du$$ or $$\int^{-\infty}_0 f(u) du$$ My question is, which of these ...
5
votes
5answers
98 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
1
vote
0answers
19 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
0
votes
1answer
33 views

Evluating triple integrals via Spherical coordinates

Use Spherical coordinates to evaluate the triple integral $$\iiint_{\mathrm{x^2+y^2+z^2<z}}\sqrt{x^{2}+y^{2}+z^{2}}\, dV,$$ What I tried Converting $x^2+y^2+z^2<z$ to Spherical coordinates ...
3
votes
0answers
406 views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
6
votes
0answers
67 views

Real analytic methods for the following integral [duplicate]

A few days back, the following integral was posted $$\int_0^1 x^x(1-x)^{1-x}\sin(\pi x)\,dx=\frac{\pi e}{24}$$ The integral was answered using complex analysis tools but I am interested in other ...
0
votes
0answers
41 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Proof:$\forall a,b;c,d\in\mathbb{R},a<b,c<d.$ $f (x+y) $ is ...
0
votes
1answer
13 views

Area between $y=2+|x-1|$ and $y=-\frac{1}{5}x+7$

Question 17, page 448, from Anton 8th. The question asks for the area, and the answer is 24. Now, I did draw the graph, found the points where both functions touch each other by $f(x) = g(x)$: ...
1
vote
1answer
44 views

A inverse Trigonometric multiple Integrals

How to calculate the closed form of the integral $$\int\limits_0^1 {\frac{{\int\limits_0^x {{{\left( {\arctan t} \right)}^2}dt} }}{{x\left( {1 + {x^2}} \right)}}} dx$$
0
votes
1answer
27 views

Is there a clever way to determine negative area of an integral?

Given some continuous, integratable function f(x) that has only positive area over a range from x1 to x2...is there a way to determine the negative area of the integral of f(x) - c (from x1 to x2), ...
8
votes
1answer
104 views

Closed-form of $\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx$

Inspired by this question, is there a closed-form of $$\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx\,?$$ Here $n \in \mathbb{N_+}$. In the answers to the question above we could find proofs of ...
7
votes
3answers
82 views

Improper integral : $\int_0^{+\infty}\frac{x\sin x}{x^2+1}$ [on hold]

How to evaluate the following improper integral : $$\int_0^{+\infty}\frac{x\sin x}{x^2+1}\,dx$$ I have tried integration by parts and variable substitution, but it didn't work.
4
votes
4answers
183 views

Inverse Trigonometric Integrals

How to calculate the value of the integrals $$\int_0^1\left(\frac{\arctan x}{x}\right)^2\,dx,$$ $$\int_0^1\left(\frac{\arctan x}{x}\right)^3\,dx $$ and $$\int_0^1\frac{\arctan^2 x\ln x}{x}\,dx?$$
3
votes
3answers
85 views

Evaluate integral: $\int_0^{+\infty}\frac{\cos{bx}-\cos{ax}}{x}dx$

It seems that $\displaystyle\int_0^{+\infty}\frac{\cos x}{x}$ is divergent, so how to solve this problem? $$\int_0^\infty \frac{\cos bx -\cos ax}{x}\, dx\quad,\quad\mbox{where}\,a,b>0$$ It's ...
1
vote
1answer
29 views

Change of variable in double integrals

I need help to solve the following question(s). a) Evaulate the integral $$\iint_D (x-y) \, dx \, dy,$$ where $D$ is the triangle with vertices $(0,0)$, $(-1,1)$ och $(4,2)$. b) Evaulate the ...
8
votes
0answers
63 views

Closed form of a difficult definite integral

I'm looking for a closed-form expression for the value of this integral: $$I=\int_0^\pi \frac{\sin(x)}{\sqrt{x^3+x+1}} dx$$ The graph of the integrand looks like this: $\hskip 2.4 in$ Numerically, ...
2
votes
5answers
641 views

How can I show that these integrals are zero

How can I show that these integrals equal $0$ when $n$ and $m$ are both integers and $n \neq m$? $$\int_{-\pi}^{\pi}\sin(mx)\sin(nx)dx = \int_{-\pi}^{\pi}\cos(mx)\cos(nx)dx = 0$$ I'm able to show that ...
7
votes
0answers
49 views

Question on the paper Donal F. Connon, “Some integrals involving the Stieltjes constants”

I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants. It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series ...
5
votes
3answers
223 views

Prove $\int_0^{2\pi}\frac{3a\sin^2\theta}{(1-a\cos \theta)^4}\mathrm{d}\theta = \int_0^{2\pi}\frac{\cos \theta}{(1-a\cos \theta)^3}\,\mathrm{d}\theta$

While doing some mathematical modelling of planetary orbits I have come up with two definite integrals $D_1$ and $D_2$ which appear to produce the same result when I test (using www.WolframAlpha.com) ...
11
votes
5answers
200 views

Integral: $\int_0^{\pi/12} \ln(\tan x)\,dx$

I am trying to evaluate: $$\int_0^{\pi/12} \ln(\tan x)\,dx$$ I think the integral is quite simple but I am having a hard time evaluating it. I started with the result: $$\int_0^{\pi/4} \ln(\tan ...
5
votes
2answers
161 views

Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$

At first sight it looks like the integral below $$\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$$ can be evaluated by using some geometric series. What else can we do? Is there a fast easy ...
1
vote
0answers
22 views

Jacobian determinant of unitary transformation

Is the Jacobian determinant of a unitary transformation equal to one? I ask because I get that impression from the appendix of this paper. They have spherical coordinates for two particles, ...
1
vote
1answer
77 views

A function $f(x)$ that Riemann integrable on $[a,b]$.

Define a function $f(x)$ that Riemann integrable on $[a,b]$. Let $$g(x)=\begin{cases} f(x)&\text{if}&x\in[a,b], \\ f(a)&\text{if}&x<a, \\ f(b)&\text{if}&x>b. ...
4
votes
2answers
64 views

An inequality with $a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$

Let the sequence $(a_n)_n$ defined by $$a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$$ 1)Prove that $$\frac12 \leq a_n \leq ...