# Tagged Questions

0answers
61 views

### Why Riemann integration is needed? [closed]

What is the necessity of the notion of "Riemann Integration" ? Why is normal definite integral is not good enough ?
0answers
51 views

### Integral involving exponential, power and Bessel function

Is there any formula for calculating the following definite integral, including exponential and Bessel function? $$\int_0^{a}x^{-1} e^{x}I_2(bx)dx$$ Thanks in advance
2answers
58 views

### Explanation for these transformations of integrals

I've recently found the following transformations: $$\int _{a} ^{\infty} \frac{\ln x}{x^2 + a^2}\,dx = \int _{a} ^{0} \frac{\ln x}{x^2 + a^2}\,dx$$ \int _{0} ^{\pi/2} ...
2answers
79 views

### Find the following integral (most likely substitution)

$$\int_0^1 \frac{\ln(1+x^2)}{1+x^2} \ dx$$ I tried letting $x^2=\tan \theta$ but it didn't work. What should I do? Please don't give full solution, just a hint and I will continue.
2answers
69 views

### Definite trig integral [duplicate]

How do I evaluate: $$\int_{0}^{\pi} \sin (\sin x) \ dx$$ I have seen a similar question here but can't find it.
3answers
94 views

### How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$

Let $f:\left[a,b\right]\to\mathbb{R}$ be a function that is derivative so that $f'$ is continuous then $$\lim_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0$$ My attempt: I ...
3answers
81 views

### Integrating a school homework question.

Show that $$\int_0^1\frac{4x-5}{\sqrt{3+2x-x^2}}dx = \frac{a\sqrt{3}+b-\pi}{6},$$ where $a$ and $b$ are constants to be found. Answer is: $$\frac{24\sqrt3-48-\pi}{6}$$ Thank you in advance!
1answer
75 views

### Evaluating the following integral:

I am trying to evaluate this integral: $$\int_{0}^{\infty }\frac{\cos(x)}{1+x^{2}}dx$$ My attempt: \int_0^{\infty}\frac{\cos(x)}{(x+i)(x-i)}dx=1/2 \int_{-\infty}^{\infty} ...
3answers
84 views

### Is it possible to evaluate this integral? [duplicate]

Is it possible to evaluate this integral: $$\int_{0}^{\frac{\pi }{2}}\ln(\sin 2x){\rm d}x$$
3answers
146 views

### Tricky elementary integral

$$\int_{0}^{\frac{\pi }{2}}x\cot(x)dx$$ I tried integration by parts and got $\frac{1}{2}\int_{0}^{\frac{\pi }{2}}x^{2} \csc^{2}x dx$ which doesn't help at all. I don't really know what to do. Any ...
0answers
48 views

### Integral of product of normal cdf and pdf

What do you think, is there a closed form solution of the following Integral $\textbf{ }$ $$\int_{-\infty}^{a-y}n(x)\, N(b-2y-x)\, dx,$$ where $N(x)=\int_{-\infty}^x n(z)\, dz\quad$ and \$\quad ...
3answers
35 views

### Confusing Triple Integral

i'm having trouble with this integral the integral is $\int_0^9\int_{\sqrt z}^3\int_0^y z\cos(y^6)\,dx\,dy\,dz$. We aren't given any more information and i'm a bit stuck as to where to start. I don't ...
2answers
53 views

### Triple integral problem involving a sphere

Let $R = \{(x,y,z)\in \textbf{R}^3 :x^2+y^2+z^2\le\pi^2\}$ How do I integrate this triple integral $$\int\int\int_R \cos x\, dxdydz,$$ where $R$ is a sphere of radius $\pi$? I have trouble ...
2answers
121 views

### Trig Fresnel Integral

$$\int_{0}^{\infty }\sin(x^{2})dx$$ I'm confused with this integral because the square is on the x, not the whole function. How can I integrate it? Thank you. I have not done complex analysis (only ...
1answer
80 views

### What is the proof that anti-derivative gives function = area under curve?

For many years now I have thought about this but have not been able to get a clear answer. We all know that $\displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ gives us a function we call as the ...
2answers
28 views

### Is it true that $\int_1^ba^{\log_b x}dx> \log_eb$

Is it true that $\int_1^ba^{\log_b x}dx> \log_eb$ $\forall a,b>0\ and\ b\not = 1$
1answer
25 views

### definite integral negative variable

Man, it's been so long since I did this. I am trying to do this: NB: limits are $-\pi$ and 0, but I can't get the minus in the limits. If anybody knows how do to that please let me know, the $\pi$ ...
3answers
86 views

### Is this definite integral impossible?

From my understanding when you integrate $f(x)$ you get $F(x)+C$, and when finding a definite integral the $C's$ cancels out due to subtraction. However, I came across an example where the $C$ doesn't ...
0answers
26 views

### Complex Fourier series of a function [duplicate]

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
2answers
181 views

### Complex Fourier series

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
3answers
95 views

### Is this integral right?

$$\pi\int_0^{x}\left(\cot(\pi t)-\frac{1}{\pi t}\right)dt=\log\frac{\sin(\pi x)}{\pi x}$$ (original image) Is this integral right? Regardless of whether it's right or not, please give me a procedure ...
2answers
86 views

### Evaluating $\int_{\mathbb{R}}\frac{\exp(-x^2)}{1+x^2}\,\mathrm{d}x$

I would like to evaluate in a closed form the integral $$\int_{\mathbb{R}}\frac{\exp(-x^2)}{1+x^2}\,\mathrm{d}x$$ I tried various methods : integration by parts some changes of variables ...
1answer
54 views

### Prove that $\int_0^x \int_0^y \int_0^z f(t) dt dz dy = \frac{1}{2} \int_0^x (x-t)^2 f(t) dt$

Prove that $$\int_0^x \int_0^y \int_0^z f(t) dt dz dy = \frac{1}{2} \int_0^x (x-t)^2 f(t) dt$$ Came across this problem and I'm not even sure how to start it. I figured that if the end goal is ...
2answers
50 views

### Integration solving problem

A integration is given $$x-x_0 = \pm \int_{0}^{\phi(x)}\frac{d\Phi}{\sqrt\frac{\lambda}{2}(\Phi^2-\frac{m^2}{\lambda})} \tag{1}$$ The author said that, equation (2) can be written from equation (1) by ...
3answers
235 views

### integral with $\log\left(\frac{x+1}{x-1}\right)$

I encountered a tough integral and I am wondering if anyone has any ideas on how to evaluate it. \displaystyle ...
2answers
161 views

### Let $f:[a,b]\to\mathbb R$ be Riemann integrable and $f>0$. Prove that $\int_a^bf>0$. (No Measure theory) [closed]

Is the Riemann integral of a strictly positive function positive? This is not a duplicate. I'm specifically interested in a proof not involving Measure Theory. The thread above uses the fact that $f$ ...
0answers
105 views

### Riemann sums vs Darboux sums

Let we speak of the tagged partitions of an interval and a bounded function defined on it. I think that the tags give rise to particular Riemann sums which may be quite different in value that the ...
3answers
77 views

### Integral with Bessel functions of the First Kind.

I'd like to solve the following integral: $I = \int_0^\infty J_0(at) J_1(bt) e^{-t} dt\$ where $J_n$ is an $n^{th}$ order Bessel Function of the First Kind and $a$ and $b$ are both positive real ...
3answers
55 views

### Integration question

I have trouble in integrating the following integral. I would appreciate any help :D $$\int_0^1 \sqrt{-\log x}\, a\, x^{a-1}dx$$ Thanks heaps :D The answer is $\sqrt{\pi}/2(\sqrt{a})$.
2answers
78 views

### Integrating by substitution

I'm embarrassed to ask this question, but what's the flaw in the following evaluation? $\displaystyle\int_{0}^{\pi} \sin (\sin x) \ dx = \int_{0}^{0} \frac{\sin u}{\sqrt{1-u^{2}}} \ du = 0$.
2answers
163 views

### Is the Riemann integral of a strictly positive function positive?

In the proof here a strictly positive function in $(0,\pi)$ is integrated over this interval and the integral is claimed as a positive number. It seems intuitively obvious as the area enclosed by a ...
0answers
56 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 ...
0answers
30 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 ...
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 ...
1answer
66 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 ...
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, ...
1answer
114 views

### Improper integral evaluation

I'm looking for a method to evaluate the following integral: $\displaystyle \int_0^{\infty} \left( \frac{1}{e^x - 1} - \frac{1}{x} + \frac{e^{-x}}{2} \right) \frac{1}{x} dx$ EDIT: Using the link, ...
3answers
61 views

### Definite Integral with a discontinuty

I have the next integral: $$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$ I have no clue how to start. At $x=0$ there is a clear discontinuity and I don't know how to solve the integral. The main ...
2answers
56 views

### Need to prove $\frac{3}{5}(2^{\frac{1}{3}}-1)\le\int_0^1\frac{x^4}{(1+x^6)^{\frac{2}{3}}}dx\le1$

I need to show that $$\frac{3}{5}(2^{\frac{1}{3}}-1)\le\int_0^1\frac{x^4}{(1+x^6)^{\frac{2}{3}}}dx\le1$$ I just know that if in $[a,b]$, $f(x)\le g(x)\le h(x)$, then ...
1answer
87 views

### Evaluating the integral $\int_0^1\arctan(1-x+x^2)dx$

I need to evaluate $$\int_0^1\arctan(1-x+x^2)dx$$ What I did: First I assume $$I=\int_0^1\arctan(1-x+x^2)dx=\int_0^1\arctan((x-\frac{1}{2})^2+\frac{3}{4})dx$$ Since the function is symmetric about ...
2answers
60 views

### Definite integral of an exponential quotient

I was wondering if someone could help me find the definite integral of this: $$\int\limits_{R1}^{R2} \frac{t\, dt}{(t^2 + K^2)^{3/2}}$$ Where $\,K,\, R1,\, R2\,$ are constants, $\,R2>R1\,$ , ...
1answer
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### A multiple integral question II

We know from the previous post that \lim_{n\to\infty}\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} ...
1answer
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### A multiple integral question

Proving that $$\lim_{n\to\infty}\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} \mathrm{d}x_1\cdot\mathrm{d}x_2\cdots\mathrm{d}x_n=1$$
2answers
54 views

### Definite integral with functions in the sides

Im trying to resolve the next definite integral: $$\int_{1-x^2}^{1+x^2}{\ln(t^2)\ dt}$$ Im not sure if I can use the Barrow's theorem, I think I have to use the fundamental theorem of integral ...
0answers
89 views

### Integral representation of Euler's constant

Prove that : $$\gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$ where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$). This integral was mentioned in Wikipedia as in ...
1answer
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### Prove that $\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$

Let $f$ , $g$ be real continuous functions in $[a,b]$. Prove that there is $c\in(a,b)$ such that $$\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$$ What would you suggest me to ...
3answers
89 views

### Integration by parts question,, possibly a circular example [duplicate]

I am having trouble figuring this out. $$\int_0^{1/3} \sec^3(\pi x) \, dx$$ We are currently doing integration by parts,, so I set $g(x)=\sec^3(\pi x)$ and $f'(x)=1$. I arrived at: x\sec^3(\pi x) ...
4answers
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### Calculate : $\int_1^{\infty} \frac{1}{x} -\sin^{-1} \frac{1}{x}\ \mathrm{d}x$

Find : $\displaystyle \int_1^{\infty} \frac{1}{x} -\sin^{-1} \frac{1}{x}\ \mathrm{d}x$. I've done some work but I've got stuck, you may try to help me continue or give me another way , in both cases ...
1answer
124 views

### Characteristic function of the Smith-Volterra-Cantor set

Let the characteristic function of the SVC set be denoted by $\beta$. Does the Riemann integral $\displaystyle \int_{0}^{1} \beta ~ d{x}$ exist? I think it does since $\beta$ is bounded, but I ...
3answers
366 views

### Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$

Many recent questions have been asked here similar to this integral $$\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x} = 2.39587\dots$$ whose "closed form" I cannot seem to figure out. I have ...
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