Tagged Questions

Questions about the evaluation of specific definite integrals.

learn more… | top users | synonyms (1)

5
votes
0answers
39 views

Evaluating by real methods $\int_0^{\pi/2} \frac{x^5}{2-\cos^2(x)}\ dx$

I'm sure you guys can briefly get the result by some methods of complex analysis, but now I'm only interested in real analysis methods of proving the result. What would you propose for that? ...
0
votes
1answer
28 views

U-Substitution. Why do you multiply the integrand by -1 in this case?

$$\int_0^{\pi/2} \! \frac{\sin x\cos x}{(4-\sin^2 x)^2} \:\text{d}x$$ set $u = 4-\sin^2 x$, therefore $du = -2 \sin x \cos x \text{d}x $ $$-\frac{1}{2} \int u^{-1/2} \text{d}u $$ Change the range ...
1
vote
1answer
49 views

Tough definite integration

For a curve given by: $x=e^{-t}\cos{2t}$, $y=\sin t$ R is the region bounding this curve, the x axis and the y axis (y-intercept is point a and x-intercept is point b). Find the exact coordinates ...
2
votes
0answers
47 views

A double integration with an embedded integral which is hard to solve

I have written this in way to make it as much as possible non-confusing. I will start describing my problem and I will walk you through my question, I have a double integration which I am trying to ...
4
votes
0answers
82 views

How to compute or simplify this nasty integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
8
votes
1answer
76 views

A limit evaluating to $2 K$ (Catalan's constant)

Experimentally I discovered the limit below that says that $$\lim_{n\to\infty} \int_0^{\pi/2} \frac{1}{\displaystyle ...
0
votes
1answer
25 views

How to find a bound for these (simple) integrals

With help of $\int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$ and $\int_{0}^{\infty} e^{-x} dx =1$, I would like to know how to derive the following bounds: $$\int_{0}^m 4e^{-\frac{t^2}{8m}}dt ...
20
votes
2answers
169 views

Prove the integral evaluates to $\frac{K}{\pi}$

Yesterday I received the following integral that might require some tedious steps to do $$\int_0^{\infty}{\small\left[ \frac{x}{\log^2\left(e^{\large x^2}-1\right)}- \frac{x}{\sqrt{e^{\large ...
0
votes
2answers
26 views

Integrated series identity with Legendre Polynomials

The Legendre Polynomials can be defined in many different ways and have several properties. Many of these can be found in books or in the net, but I couldn't find this one anywhere: Prove that: ...
0
votes
0answers
25 views

Help in evaluating the integral at the given limits

Hi guys I am hoping to get some help here. The indefinite integral below gives the following result. $$\int \left(\frac{0.0016 \left(1-\exp \left(-0.0112 v^{0.25}\right)\right)}{v^{0.5}}+\frac{0.0036 ...
0
votes
1answer
26 views

laplace method on this integral

How to get the leading asymptotic expansion for this integral $\int_{0}^{\pi/2}\sqrt{\sin(t)}\exp(-x\sin^4(t))dt $ in the limit $x\rightarrow\infty$ ? Because the maximum of the exponent is at $t=0$ ...
6
votes
3answers
73 views

Finding the integral $\int_{0}^{\infty }\frac{1}{(4x-3)(4x-1)}\,dx$.

Which method that will be effective for solving this integral?
3
votes
0answers
88 views

$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$

$$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$$ i.e. an oscillation with frequency $3\Im(a)t^2 + 2\Im(b)t + \Im(c)$ and phase $0$, multiplied ...
3
votes
2answers
63 views

Evaluation of $\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \mathrm{d}x$

I have got a trouble with integral $$\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \, \mathrm{d}x,\quad |\alpha|<1.$$ My teacher said there are two ways of solving such ones, if there is ...
2
votes
1answer
28 views

Trigonometric Integrals times exponential

Let's say I want to evaluate the integral: $$\int_0^{\pi/2} e^{ax}\cos^{a}(x) \,dx$$ where $1\leq a \leq 5$ . One standard way to go around would be by applying parts. That would result of course in ...
5
votes
5answers
182 views

Intuitive explanation of integral identity

I have been able to prove the identity $$\int_{0}^{1} \frac{f(x)}{f(x)+f(1-x)} \, dx = \frac{1}{2}$$ for any continous $f:[0,1]\to[0,\infty)$ for which the integrand is defined, with calculus, but I ...
1
vote
3answers
73 views

Solve $ \int_0^{\sqrt{\pi / 2}}\left(\int_x^{\sqrt{\pi / 2} }\sin(y^2) dy \right)dx$

I'm trying to solve this: $$ \int_0^{\large\sqrt{\frac{\pi}{2}}}\left(\int_x^{\large\sqrt{\frac{\pi}{2}}}\, \sin y^2\, dy \right)dx $$ But I'm having trouble with finding an primitive to $\sin(y^2)$. ...
0
votes
0answers
22 views

Does it really matter that we are using the Taylor polynomial and remainder?

Assuming that the quadrature rule $I_n$ integrates all polynomials of degree less than or equal to N exactly: $I_n(p)$=$I(p)$ for all p $\epsilon$ $P_N$. Using this it could be proved that for any ...
0
votes
2answers
26 views

integrate sine at denominator

This integral: $\int_{\pi/2}^0\frac{d\theta}{1-\gamma\sin 2\theta}$. I tried $e^{i\theta}=\cos\theta+i\sin\theta$ and $\sin^2\theta+\cos^2\theta=1$, but didn't succeed. Is there any one can help me? ...
1
vote
1answer
49 views

Help in computing this integration?

Any thoughts or hints on solving the following integral $$ \int_{y}^{+\infty} \frac{ e^{-\sqrt{v}} }{1+ s v^{-1} } dv $$ and where $$s= \frac{2}{x^{-1}+y^{-1}}$$ The result should be a function ...
9
votes
3answers
80 views

How to prove $\int_0^{2\pi} \ln(1+a^2+2a\cos x)\, dx=0$.

How can I prove $\int_0^{2\pi} \ln(1+a^2+2a\cos x)\, dx=0$, where $a<1$? Thanks.
0
votes
2answers
31 views

Help with the integral $\int_{-b/2}^{\frac{\pi-10b}{20}}\frac{\mathrm{d}x}{x^2+bx+c}$

Let $b$ and $c$ satisfy the equation $4c-b^2 = \frac{\pi^2}{100}$, then how should I solve the integral: $I =\int_{-b/2}^{\frac{\pi-10b}{20}}\frac{\mathrm{d}x}{x^2+bx+c}$ All I know is that the ...
10
votes
2answers
89 views

Evaluating $\int_0^{\infty}\left(\frac{\ln(1+x^2)}{1+x^2}\right)^3 dx$

I’m looking for a closed form of this integral $$I_3=\int_0^{\infty}\left(\frac{\ln(1+x^2)}{1+x^2}\right)^3 dx$$ I’ve managed to evaluate $$I_1=\int_0^{\infty} \frac{\ln(1+x^2)}{1+x^2}dx=\pi \ln 2$$ ...
0
votes
0answers
22 views

Finding the inverse of an integral

I'm looking for a computational approach here, since I don't think there is a closed-form solution. I have the following: $$ s(x) = \rho + \int_{\rho}^{x} \sqrt{ 1 + (\alpha \cos t - k)^2 } \, dt $$ ...
4
votes
1answer
71 views

integral of a function

I wanted to find the integral of the function $f(x)$ from zero to one: $$f(x)=\begin{cases}2x\sin(1/x)-\cos(1/x) & : x\in(0,1]\\ 0 & :x=0\end{cases}$$ but I think whether its integral is not ...
4
votes
1answer
34 views

Three integral involving polylogarithm function

$\newcommand{\Li}{\operatorname{Li}}$Evaluate the following integrals $$\int\limits_0^1 \frac{\Li_2^3(x)}{x}dx, \quad \int\limits_0^1 \frac{\Li_2^2(x)\Li_3 (x)}{x} dx, \quad \int\limits_0^1 ...
2
votes
1answer
33 views

Integration of complex functions

Show that $\displaystyle \int_{-\infty}^\infty x^{2n}e^{-x^2}dx=(2n)!\frac{\sqrt\pi}{4^n n!}$ by differentiating the equation $\displaystyle \int_{-\infty}^\infty e^{-tx^2} dx=\sqrt{\frac{\pi}{t}}$. ...
0
votes
1answer
46 views

rational exponent of negative base

I have the definite integral $$\int_{1}^{\,9} {\frac{6}{\sqrt[3]{x-9}}}\, \mathrm dx$$ When I try to evaluate it I get the indefinite integral equals $9(x-9)^{2/3}$ and evaluating at the limits gives ...
2
votes
1answer
147 views

How to evaluate $ \int_{0}^{1} x^{x^{x^{x^…}}} dx $

Inspired by the fact that $\int_0^1 \frac{1}{x^x}=\sum_{k=1}^\infty \frac{1}{k^k}$ I asked myself wether it is possible to evaluate the following integral: $$ \int_{0}^{1} x^{x^{x^{x^…}}} dx $$ In a ...
9
votes
2answers
175 views

Closed form of $\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$

What real analysis tools would you recommend me for getting the closed form of the integral below? $$\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$$
7
votes
2answers
145 views

Using differentiation under integral sign to calculate a definite integral

I want to calculate the integral $$\int^{\pi/2}_0\frac{\log(1+\sin\phi)}{\sin\phi}d\phi$$ using differentiation with respect to parameter in the integral ...
-4
votes
0answers
57 views

Show $\int_0^{\pi/2}\ln\biggl(\frac{\ln^2(\sin x)}{\pi^2+\ln^2(\sin x)}\biggr)\frac{\ln(\cos x)}{\tan(x)}dx=\frac{\pi^2}{4}$ [duplicate]

Yesterday I found this integral on Quora: How does one prove the following integral? $$ \int_0^{\pi/2}\ln\biggl(\frac{\ln^2(\sin x)}{\pi^2+\ln^2(\sin x)}\biggr)\frac{\ln(\cos ...
4
votes
3answers
121 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
95 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
26 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). ...
14
votes
1answer
136 views
0
votes
0answers
24 views

volumes and integrals help [closed]

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
38 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} ...
8
votes
3answers
112 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 ...
4
votes
1answer
117 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
47 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
33 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 ...
13
votes
6answers
241 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 ...
3
votes
0answers
84 views
+150

Closed formula for the 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 ($a$ and $b$ are positive constants). ...
0
votes
0answers
41 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
99 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 ...
1
vote
1answer
27 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
46 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
92 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?