Questions about the evaluation of specific definite integrals.
2
votes
1answer
98 views
Finding the limit $\lim\limits_{n \to \infty} \int_1^a \frac{n}{1+x^n} dx$ with $a > 1$
$I_n$ is given by $$I_n=\int_1^a \dfrac{n \ dx}{1+x^n}, \qquad a>1.$$
Find $\lim\limits_{n\to\infty}I_n$.
0
votes
1answer
66 views
Contour Integrals
Evaluate:
$\int_C \hat{z} dz$ where $C$ is the straight line from $i$ to $2-i$.
$\int_C \frac{dz}{z}$ where $C$ is the straight line from $3$ to $4i$
$\int_C (z-z_0)^{n-1}dz $ for any integer $n$, ...
8
votes
2answers
159 views
Evaluating $\int_0^{2\pi} \sqrt{1+\cos^2(x)}\ dx$ [duplicate]
I want to evaluate this integral
$$\int_0^{2\pi} \sqrt{1+\cos^2(x)}\ dx$$
But I cannot find a useful substitution/strategy. Could you please give me a hint?
I was thinking proving that this is ...
3
votes
1answer
104 views
About $\int_0^{\pi/2}\arctan(1-\sin^2 (x) \cos^2 (x))dx = \pi \left( \frac{\pi}{4}-\pi \arctan \sqrt{\frac{\sqrt{2}-1}{2}}\right)$
In this question sos440 has mentioned about an integral that he computed:
$$\int_0^{\pi/2}\arctan(1-\sin^2 (x) \cos^2 (x))dx = \pi \left( \frac{\pi}{4}- \arctan \sqrt{\frac{\sqrt{2}-1}{2}}\right)$$
...
3
votes
0answers
58 views
Simplify the integral with error function
$\newcommand{\erf}{\operatorname{erf}}$
I have the following integral and I need to simplify the solution. I have written first two steps. I don't know what is the value of
$$
\erf(\infty)
$$
I ...
8
votes
1answer
154 views
Interesting log sine integrals $\int_0^{\pi/3}x \log^2 \left(2\sin \frac{x}{2} \right)dx= \frac{17\pi^4}{6480}$
Show that
$$\begin{aligned} \int_0^{\pi/3}x \log^2 \left(2\sin \frac{x}{2}
\right)dx &= \frac{17\pi^4}{6480} \\ \int_0^{\pi/3}\log^2
\left(2\sin\frac{x}{2} \right)dx &= ...
0
votes
1answer
105 views
Delta (Dirac) function integral
I have a question about the integral of the $\delta$ (Dirac) function.
Is the following correct?
$\int^{c}_{0} d\delta(x) = -\delta(0) = -\infty$
where c>0.
Can we have a specific number for the ...
3
votes
0answers
79 views
What is $\int_0^{\infty}\!e^{-x^2}e^{-ae^{bx^2}}\,dx$?
I've been trying without success to evaluate
$$
\int_0^{\infty}\!e^{-x^2}e^{-ae^{bx^2}}\,dx.
$$
It's not in my integral tables. Wolfram online integrator won't do it. It doesn't seem to be amenable to ...
0
votes
1answer
68 views
$\int_{0}^{\pi/2} (\sin x)^{1+\sqrt2} dx$ and $\int_{0}^{\pi/2} (\sin x)^{\sqrt2\space-1} dx $
How do I evaluate $$\int_{0}^{\pi/2} (\sin x)^{1+\sqrt2} dx\quad \text{ and }\quad \int_{0}^{\pi/2} (\sin x)^{\sqrt2\space-1} dx \quad ?$$
4
votes
1answer
38 views
Show that $\int_{0}^{\frac \pi2} f(x) \operatorname{d}x = \pi/4$
$f(x) = \dfrac{\sin^\alpha x \operatorname{d}x}{\sin^\alpha x + \cos^\alpha x}$
Well, if $\alpha $ = 0,
$$ \int_{0}^{\frac \pi2} \frac 12 dx = \frac 12 \frac \pi2 = \frac \pi4$$
But I don't know ...
0
votes
0answers
31 views
Solving an complex Integration with complex exp and other terms
I am trying to solve a partial differential equation and while solving I need to solve the following integral. If anyone could help me solve this integral that would be great.
$$y(x,t) = \int_{c-i ...
1
vote
0answers
44 views
Series sum and integral inaccuracy problem
Well, it happened that math is not my field. Having a vague concept in general I have been struggling for a few hours with the exercise. In general, I want to calculate how much interest a man have ...
0
votes
2answers
51 views
definition of a graph
I have two questions;
What is the name of the graph (or circuit) which goes along the outer vertices of existing nodes.
What will be the formal definition of that graph.
for the ...
4
votes
2answers
74 views
Show $\int_0^b \sqrt{\frac{1 + \cos \theta}{\cos \theta - \cos b}} d\theta= \pi$ for $0 < b < \pi.$
I came across this integral in evaluating the time a particle takes to travel
between points on the cycloid. I was able to use substitution to do the integral in the case $b = \pi/2.$ I have not not ...
1
vote
1answer
118 views
Improper integral evaluates to $-\pi^2/12$
$$\int_0^1 \frac{\ln x}{1+x} \mathrm{d}x=-\int_0^1 \frac{\ln(1+x)}{x}\mathrm{d}x=-\frac{\pi^2}{12}$$
Please, help give me proper hints to solve.
I was not even able to equate the first two.
0
votes
0answers
49 views
Integrating and Triangle after Transformation to a standard simplex
I am new to this forum and even don't know how to write these mathematical symbols. I have a question below and your reply would be highly appreciated.
If I have a triangle with vertices $V_1 = ...
3
votes
0answers
179 views
Finding the volume of a tetrahedron by given vertices.
Please help me with the problem below.
Find the volume of a tetrahedron with vertices:
$O(0,0,0)$, $A(1,2,3)$, $B(-2,1,5)$, $C(3,7,1)$ by using triple integral.
Hint: First find the the equations of ...
0
votes
2answers
40 views
Show that the integrals are equivalent
Show that:
$$\int_o^{\infty}\frac{\cos(x)}{1+x}dx=\int_o^{\infty}\frac{\sin(x)}{(1+x)^2}dx$$
I have no idea how to approach. The only thing I can think is substitution $y=\pi/2-x$ or integration by ...
1
vote
2answers
58 views
Evaluate a big expression.
If:$$\lambda=\int_{0}^{1}\frac{dx}{1+x^3} ;$$
Then evaluate : $$ p=\lim_{n \rightarrow \infty}\left( \frac{\prod_{r=1}^{n}\left(n^3+r^3\right)}{n^{3n}} \right)^{1/n} ;$$
EDIT: I have now ended upto ...
1
vote
1answer
77 views
How to find limits of integration on a convolution of CRVs
In finding the convolution of two independent and continuous random variables, I am struggling with limits of integration. I cannot seem to figure out over what intervals the probability density ...
7
votes
2answers
88 views
Evaluation of $\int^1_0\frac{x^b-x^a}{\ln x} d x=\ln{{\frac{1+b}{1+a}}}$
$$\int^1_0\frac{x^b-x^a}{\ln x} d x =\ln{{\frac{1+b}{1+a}}}$$ The part inside $\ln$ is absolute value. A solution including integration under the integral can be found here. Which used the identity ...
1
vote
0answers
57 views
Mellin tranform of $\cos x$ using Ramanujan's master theorem
I've been messing around with Ramanujan's master theorem.
$\displaystyle \int_{0}^{\infty} x^{s-1} \cos x \ dx = \int_{0}^{\infty} x^{s-1} {}_0F_1 \left(-;\frac{1}{2};\frac{-x^{2}}{4} \right) \ dx $
...
5
votes
1answer
81 views
Calculate $\int_0^1 \frac{-\ln(1-x)}{x} d x$ without $\zeta(2)$.
Can we calculate the following integral without the need of $\zeta(2)$, I actually believe that this can be a method to find the accurate value of $\sum_{n\geq 1}n^{-2}$.
$$\int_0^1 -\frac{\log ...
13
votes
0answers
205 views
$\int_0^1\mathrm{d} u_1 \cdots \int_0^1\mathrm{d} u_n \frac{\delta(1-u_1-\cdots-u_n)}{(u_1+u_2)(u_2+u_3)\cdots(u_{n-1}+u_n)(u_n+u_1)}$
This question grew out of this one: Given an even integer $n\in 2\mathbb{N}$, compute the integral
$$\int_0^1\mathrm{d} u_1 \cdots \int_0^1 \mathrm{d} u_n ...
0
votes
1answer
88 views
Getting weird integral evaluation from Wolfram Alpha
Check this out. I hand-evaluated this integral and my pretty sure the answer is zero, but Wolfram returns the value $4i\pi$ instead.
0
votes
2answers
30 views
Simple contour intergal along a circular path
Am I correct that the the following integral evaluates to 0 since the domain of integration is a closed loop and the integrand is continuous over the loop?
$$\int_{C(0,7)}\frac 1{(z-1)(z-3)} dz$$
4
votes
3answers
194 views
How to calculate volume of $z = 0, z = 1, x+y+z=2, x = 0, y = 0$ area by tripple integral?
I am calculating volume of body that is defined by $z = 0, z = 1, x+y+z=2, x = 0, y = 0$ to do this I have two possible ways:
...
0
votes
1answer
85 views
Parameterize the following about a unit circle
How do I compute the following integral using contour integration:
$$\int_0^{2\pi} \frac{\cos^2 \theta}{6-2\cos\theta}d\theta$$
9
votes
2answers
195 views
How to find integral of $\int_0^\infty \frac{\ln ^2z} {1+z^2}\mathrm{d}z$?
How do I find the value of $$\int_{0}^{\infty} \frac{(\ln z)^2}{1+z^2}\mathrm{d}z$$ without using contour integration, - using the usual special functions, e.g., zeta/gamma/beta/etc.
Thank you,
1
vote
1answer
42 views
How to show this function is not in $L^{p}$ for any $p \neq 2$?
For $E = (1, \infty)$ and $f$ defined by $$f(x) = \frac{x^{-1/2}}{1 + \ln{x}} \text{for } x >1$$ $f$ belongs to $L^{p}{(E)}$ iff $p = 2$
This is an example on page 143 of Real Analysis, Royden ...
3
votes
2answers
166 views
How to integrate $1/(x^3 + 8)$?
I need some suggestions to solve this integral:
$$\int_{1}^{3} \frac{1}{x^3 + 8} dx$$
Thanks.
4
votes
0answers
56 views
Proving $\int_{0}^{\pi/2}(\log(\sin x))^2dx = \frac{1}{24}\cdot(\pi^3 + 12\pi(\log 2)^2)$ [duplicate]
Proving
$$\int_{0}^{\pi/2}(\log(\sin x))^2dx = \frac{1}{24}\cdot(\pi^3 + 12\pi(\log 2)^2)$$
without making use of gamma function,digamma function, hypergeometric function.
9
votes
1answer
96 views
Evaluating a slow sum
In my integration adventures, I came across this sum which I could not simplify:
$$\sum_{n=1}^{\infty}\frac{(-1)^{n}\log(2n+1)}{2n+1}$$
Wolfram seems to believe the sum diverges and is not of much ...
10
votes
2answers
190 views
How to calculate $\int_{0}^{1}(\arcsin{x})(\sin{\frac{\pi}{2}x})dx$?
How to find the follwing integral's value ?
$$\int_{0}^{1}(\arcsin{x})(\sin{\frac{\pi}{2}x})dx$$
Actually, I don't know it can be represented as closed form.
3
votes
2answers
104 views
How to evaluate $\int_{0}^{\pi }\frac{x\sin x}{1+\cos^{2}x}dx$ [duplicate]
How can I evaluate this integral?
$$\int_{0}^{\pi }\frac{x\sin x}{1+\cos^{2}x}dx$$
7
votes
3answers
237 views
How to do integral $\int_{-\infty}^{\infty} x^4 e^{-x^2/2}dx$
Can someone show me a simple way to do integral
$\int_{-\infty}^{\infty} x^4 e^{-x^2/2}dx$?
I am working on something related to the moments of normal distribution and require the evaluation of the ...
0
votes
0answers
30 views
Fundamental theorem of calculus 1 where integrand is a 2nd order partial derivative
I have a function $b(x,y)$ such that $b(x,0)=0$.
Now, suppose I wish to evaluate the following integral: (Note that $b$ is continuous almost everywhere but it is assumed that it is integrable. Also, ...
1
vote
1answer
186 views
Laplace transform of Bessel function of the first kind
I can't figure out why my evaluation of $\displaystyle \int_{0}^{\infty} J_{n}(bx) e^{-ax} \ dx \ (a,b >0, \ n=0,1,2, \ldots)$ if off by a factor of $b$.
$ \displaystyle\int_{0}^{\infty} J_{n}(bx) ...
1
vote
0answers
57 views
Double integral
I want to evaluate
$$ \int_{-\infty}^a \int_{-\infty}^b z_1^n z_2^k \phi(z_1,z_2) dz_1 dz_2$$
where $ \phi(z_1,z_2)$ is a pdf of bivariate normal distribution and $n,k$ are natural numbers. Are there ...
11
votes
3answers
352 views
The equivalence between Cauchy integral and Riemann integral for bounded functions
Definitions
Suppose $P\colon a=x_0<x_1<\dotsb<x_n=b$ is a partition of $[a,b]$. Let $\Delta x_k=x_k-x_{k-1}$ and $\lVert P\rVert$ denotes $\max_{0<k\le n}\Delta x_k$.
The Cauchy integral ...
2
votes
1answer
116 views
Covariance of Student's t-distribution
Another integral (this time it looks like a lot of work but maybe it can be simplified).
I have the Student's t-distribution
$$\int_{-\infty}^\infty ...
3
votes
2answers
93 views
Evaluate the integral $I=\int\limits_{-\infty}^{\infty} \frac{\sin^{2}u}{u^{2}}du$
To evaluate this integral I have got to use Parseval's Theorem and the fourier transform of
$$s(x)=\begin{cases}
0 & x\leq -a \\
1 & -a<x<a \\
0 & x \geq a.\end{cases}$$
This ...
7
votes
2answers
272 views
Definite Integral $\int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)}dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right)$
I want to prove that
$$\int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)}dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right)$$
2
votes
1answer
59 views
Integral solution of a differential equation
I'm having difficulty verifying that $$K(x)=\int_{1}^{\infty}\frac{e^{-xt}}{\sqrt{t^{2}-1}}\, dt$$
satisfies the differential equation ...
1
vote
1answer
63 views
Solving $\int_0^\infty \left(1+\frac{y^{2}_{1}+y^{2}_{2}+\cdots+y^{2}_{n})}{\nu}\right)\mathrm dy_{1}\mathrm dy_{2}\cdots \mathrm dy_{n}$
Interesting integral here:
$$\int_0^\infty \frac{\Gamma(D/2+\nu/2)}{\Gamma(\nu/2)}\frac{|\Lambda|^{1/2}}{(\pi \nu)^{D/2}}\left(1+\frac{(x-\mu)^{T}\Lambda(x-\mu)}{\nu}\right)^{-D/2-\nu/2}\mathrm dx$$
...
10
votes
1answer
121 views
Evaluate $ \int_{0}^{1}{\frac{\ln{x}}{x^2-x-1}dx}$
How can we evaluate the definite integral
$$\displaystyle \int_{0}^{1}{\frac{\ln{x}}{x^2-x-1}dx}$$
I tried many times but still have no idea.
1
vote
2answers
99 views
about $\int \cos^q(x)\sin^p(x) dx$
for all $n$ and $m$ from $\mathbb{N}$ we define $H_{m,n}$ by :
$$H_{m,n}=\int_{0}^{\frac{\pi }{2}}\cos ^{m}(x)\sin^{n} (x)dx$$
find relation between $H_{m,n}$ and $H_{m,n-2}$
find relation between ...
8
votes
2answers
196 views
Integrating $\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$
Could someone help with the following integration:
$$\int^0_\pi \frac{x \sin x}{1+\cos^2 x}$$
So far I have done the following, but I am stuck:
I denoted $ y=-\cos x $ then:
...
7
votes
2answers
271 views
Integral $\int_0^\pi \cot(x/2)\sin(nx)\,dx$
It seems that $$\int_0^\pi \cot(x/2)\sin(nx)\,dx=\pi$$ for all positive integers $n$.
But I have trouble proving it. Anyone?
0
votes
1answer
32 views
Help with integral (using integrating by part)
I need help understanding the equality under the red question mark:
Can you please be as descriptive as possible? :D
