Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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0
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1answer
14 views

Improper integral convergence question

Prove that the following integral converges: We divided the integral to 2 integrals (one from 0 to 1/2 and the other from 1/2 to 1). We managed to prove that the integral from 1/2 to 1 converges ...
0
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1answer
20 views

Is there any way to combine the product of two univariate integrals into a single integral?

Can two separate integrals but multiplied together in the end by integrated as a product once instead? In other words, does $$ \left(\int_{-\infty}^{+\infty}f(x)\mathrm{d}x ...
0
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2answers
58 views

Improper rational/trig integral comes out to $\pi/e$

During my studying to integration I find this integration. So I tried to prove but I got stuk. So I need help in this integration. $$\displaystyle\int_{-\infty}^{\infty} \frac{x \sin (x)}{1+x^2} ...
1
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2answers
44 views

Compute the integral

Compute the integral: $$\frac{1}{2\pi i}\int_{|z| = 1}\frac{(z-b)^m}{(z-a)^n}dz$$ where $|a| < 1 < |b|$; $m, n \in \mathbb{Z}$ My approach is using Cauchy integral, we have $$\frac{1}{2\pi ...
-4
votes
2answers
35 views

How we can prove this??? [on hold]

I study integration and find this... need to know this integration How come? $\displaystyle\int_{0}^{\infty} \frac{e^{-x} - e^{ -2x}}{x} dx = ln (2) $
3
votes
2answers
34 views

compute the integral $\int_{|z|=1}\left[\frac{z-2}{2z-1}\right]^3dz$

Compute this integral: $$\int_{|z|=1}\left[\frac{z-2}{2z-1}\right]^3dz$$ my solution is I used derivative of Cauchy integral formula, which is $$f^{(n)}(z_0) = \frac{n!}{2\pi i}\int ...
3
votes
3answers
132 views

Definite integral with limits from zero to infinity

Let $ I=\int\limits_{0}^{\infty}e^{-(x^2+\frac{1}{x^2})}dx$ and $J=\int\limits_{0}^{\infty}x^2e^{-(x^2+\frac{1}{x^2})}dx$. If $J=\dfrac{pI}{q}$, then find the value of $p+q$ where $p$ and $q$ are ...
4
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2answers
44 views

A problem related to integration in $L^1$

If $f\in L^1[0, 1]$ and $\int_{0}^1 x^nf(x)=0$ for all $n = 0,1,2,...$then prove that $f$ is identically zero almost everywhere. This would be very easier to prove if $f$ were continuous on $[0, 1]$ ...
2
votes
2answers
121 views

Not the toughest integral, not the easiest one

Perhaps it's not amongst the toughest integrals, but it's interesting to try to find an elegant approach for the integral $$I_1=\int_0^1 \frac{\log (x)}{\sqrt{x (x+1)}} \, dx$$ $$=4 ...
1
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4answers
63 views

Evaluate $\iint dy\,dx;\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4};0\leq r\leq2$

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4}\;\;\;\;,0\leq r\leq2$ $\color{blue}{\text{without using polar coordinates}}$. My attempt: ...
2
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0answers
41 views

Can $\int_{0}^{1}\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx$ be expressed in a simple form?

I was browsing the book Irresistible Integrals and found this gem, at page 97, $$ \int_{0}^{1}x^{n}\ln^{k}(x)dx=\frac{(-1)^{k}k!}{(n+1)^{k+1}} $$ that resembles a previous question of mine here. So, ...
1
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0answers
35 views

Integral with Bessel Functions

Any suggestions how to solve this: \begin{equation} \int_0^a\int_0^\infty J_0 (\lambda r)J_1(\lambda a)\frac{1}{\sqrt{n+\lambda^2 }}d\lambda dr \end{equation} (J0,J1 Bessel function of first kind, ...
2
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4answers
37 views

Use comparison test to determine convergence

$$\int_{1}^{\infty}\frac{\ln x}{\sinh x}dx$$ I tried several functions and failed to get integrable convergent bigger function. Thanks for help.
0
votes
1answer
30 views

Evaluate $\int_{-2}^{2}\int_{y^2-3}^{5-y^2}dxdy$ [duplicate]

In the black I evaluated the integral and I got 64/3, now I need to evaluate the same integral with $\color{red}{dydx}$ .in the $\color{blue}{\text{blue}}$ color is my attempt, I don't think that my ...
-4
votes
0answers
22 views

integrate very long expression using orthogonality in maple [on hold]

I have very long expression and i must integrate it. i try to apply "orthogonality" on my equations to eliminate "X" and "Y" variables. Image Shows examples of Orthogonal Functions my integral code ...
2
votes
2answers
38 views

A question on use of square integrable functions

I'm approaching this from a physicist's perspective, so apologies for any inaccuracies (and lack of rigour). As far as I understand it, a square-integrable function $f(x)$ satisfies the condition ...
2
votes
1answer
15 views

Properties of unimodal functions

A probability density function $f$ is said to be unimodal if the value at which the maximum value of the function is attained is unique. I am reading some papers on econometrics that appear to use ...
1
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0answers
22 views

Continuity of improper integrals

There is a theorem saying that if $f:[a,b]\to \mathbb R$ is integrable on $[a,b]$, then $F(x):=\int_{a}^{x}f(t)dt$ is continuous on $[a,b], x \in [a,b]$. Is there an analogous theorem of the kind: ...
0
votes
2answers
40 views

How to prove that $R[a,b]$ is dense in $L^1[a,b]$ ?

How to prove that $R[a,b]$ is dense in $L^1[a,b]$ ? ( where $R[a,b]$ is the set of all riemann integrable functions on $[a,b]$ )
3
votes
5answers
151 views

How to integrate $\int \frac{4}{x\sqrt{x^2-1}}dx$

In order to solve the following integral: $$\int \frac{4}{x\sqrt{x^2-1}}dx$$ I tried different things such as getting $u = x^2 + 1$, $u=x^2$ but it seems that it does not work. I also tried moving ...
-1
votes
1answer
42 views

How to find the antiderivative of f(x). [on hold]

While studying, I learned that the antiderivative of $1/f(x)$ is simply ln$\lvert f(x)\rvert$. Why is this so?
2
votes
1answer
44 views

Help on finding the closed form of the integral

Can anyone help me to find closed solution of the integral $$\int_0^{1-e^{-\lambda x}}\frac{u^{b-1}\,(1-u)^{a+c-1}}{[1-(1-e^{-\lambda_1 t_1})u]^{a+b+c}}\,{\rm d}u,$$ where ...
1
vote
1answer
41 views

Integrals, intermediate value theorem question

f∈c[a,b] (f is continuous in [a,b]), prove: We tried to use the integral intermediate value theorem to try to prove it but we don't understand why the limit has to be the max and not any other value ...
-1
votes
3answers
66 views

Integral for cos [on hold]

$$\int_{-a}^{+a}A^2\cos^2\left(\frac{n\pi a}{2}x\right)dx$$ where $A, n, \pi,a$ are constants. I was thinking about trigonometric formula for $\cos^2x=\frac{1+\cos2x}{2}$
1
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1answer
27 views

Area surrounded by a curve

I would need help to calculate the area surrounded by a curve. The curve is given with the following polar coordinates: I know we need need to integrate with respect to r and theta but am stuck ...
2
votes
0answers
28 views

Methods to Minimize Functions and Integrals over $\mathbb{N}$.

In a paper I'm writing, I have to minimize a messy function $f(\mu,n)$ where $\mu \in \mathbb{R}$ and $n \in \mathbb{N}$. That is, given $\mu \in \mathbb{R}$, I need to minimize the one variable ...
3
votes
4answers
93 views

Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\frac{3}{8}{(\frac{\pi}{\alpha^5})}^\frac{1}{2}$ [duplicate]

In my physics course this standard formula is used a lot without proof so it would be interesting to see a neat proof for it. From a previous thread by me I know the proof for $\int ...
0
votes
1answer
23 views

Why is $F'(x) = 2x·\tan(x^2)-\tan x$ if $F(x) = \int_{x}^{x^2}\tan u\, \mathrm du$?

Evaluate $F'(x)$ if $$F(x) = \int_{x}^{x^2}\tan u\, \mathrm du$$ I tried to do this by the change of variables formula and hence, $$F(x) = \int_{x}^{x^2}\tan u\, \mathrm du=\int_{\sqrt x}^{x}\tan ...
1
vote
1answer
71 views

Antiderivative of $\frac{e^x}{\sqrt{1-x^2}}$

Can anyone help me find the following indefinite integral: $$\int{\frac{e^x}{\sqrt{1-x^2}} dx}$$ I cannot think of any transformation...
0
votes
1answer
52 views

Definite integral: $\int^\pi_0 e^{2a \cos x} \left( \frac{\sin^2 x}{1- \cos x} \right) dx$

The goal is to solve this: $$ \int^\pi_0 e^{2a \cos x} \left( \frac{\sin^2 x}{1- \cos x} \right) dx $$ with $a>0$. Really not sure how to attack this one. The integrand seems to be capable of ...
2
votes
5answers
57 views

Show that $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2}$ + Constant

I tried to do this integration by parts and got $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2} +\alpha\int x^3\mathrm{e}^{-\alpha x^2}\mathrm dx$ + constant. ...
3
votes
1answer
24 views

How to prove define integrate from f(sin x)

i need help for prove this problem , i dont have idea for this prove, i very appreciate your sugerences. $$ \int ^{\pi }_{0}xf(\sin x)\,dx = \int ^{\pi }_{0}\frac{\pi }{2} f(\sin x)\,dx $$
0
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0answers
23 views

Interpretation of integral as ratio of joint and conditional densities?

A common exercise in Bayesian statistics is specifying a prior $p(\theta)$ on some parameter $\theta$. We then observe a collection of data $D=(X_1,\dots,X_N)$, the distribution of which is ...
5
votes
1answer
57 views

Derivation of Gradshteyn and Ryzhik integral 3.876.1 (in question)

In the Gradshteyn and Ryzhik Table of Integrals, the following integral appears (3.876.1, page 486 in the 8th edition): \begin{equation} \int_0^{\infty} \frac{\sin (p \sqrt{x^2 + a^2})}{\sqrt{x^2 + ...
0
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0answers
36 views

why do we need integration in the probability or measure theory? [on hold]

As the title is, why do we need integration in the probability or measure theory? because we do not learn how to calculate an area or volume under some function in the field. Some practical examples ...
4
votes
4answers
125 views

Find $\int_0^1(\ln x)^n\hspace{1mm}dx$

I am not a big fan of induction, it's just a personal preference. Is there a method other than induction. Answer is $n!$ by the way
3
votes
0answers
59 views

Calculating in closed form $\int_0^{\infty} \frac{\text{PolyLog}^{(1,0)}(1,-x)}{1+x^2} \, dx$

Can you confirm the following result? Mathematica and other computational stuff I used seem unable to do anything about this result. Maybe to confirm it numerically? $$\int_0^{\infty} ...
6
votes
0answers
72 views

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
1
vote
1answer
42 views

Easy method to check integrability as elementary functions

What could be an easy method (Calc 1) to check if a given integral is not integrably in terms of elementary functions? Take for example: $$ \int e^{-t^{2}}dt$$
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0answers
91 views

Conjecturing the closed form $\frac{\pi ^2}{8}-\frac{\pi ^2}{8 \sqrt{2}}+\frac{\pi \log (2)}{4 \sqrt{2}}$

I conjecture that $$\small \int_0^{\pi/2} \frac{\cos ^2(x) \left(-2 \log \left(4^{-\sin ^2(x)} \sin ^{-4 \sin ^2(x)}(x)\right)-4 \log (\cos (x))+\cos (2 x) (4 \log (\cos (x))+\pi +\log ...
-4
votes
0answers
50 views

How to evaluate the integral $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$? [on hold]

How to evaluate the integral $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$?
1
vote
2answers
62 views

Volume of Solid Enclosed by an Equation

I'm having problems finding the triple integrals of equations. I guess it has to do with the geometry. Can someone solve the two questions below elaborately such that I can comprehend this triple ...
10
votes
1answer
506 views

The word “integral” in calculus unrelated to “integral” / “integer” in algebra?

I think that the word integral in calculus is nothing to do with integer or integer numbers. But why is integral is chosen for integration? In algebra, integral means related to integers, and this is ...
1
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2answers
46 views

How to evaluate the line integral $\int_C (y-z)\,dx+(z-x)\,dy+(x-y)\,dz$

How to evaluate the line integral $\int_C (y-z)\,dx(z-x)\,dy(x-y)\,dz$. The curve $C$ is the intersection of the cylinder $x^2+y^2=1$ and the plane $x-z=1$. I am really stuck on how to to do this ...
0
votes
1answer
34 views

How to solve integral with natural logarithm and product

I am trying to solve the following integral: $$\int{\frac{x}{4} \ln\left(\frac{4}{x}\right)}$$ Using this integral table, the more close case is (43). However, this is not the right one to use. Do ...
1
vote
4answers
36 views

Find equation of curve

${dy \over dx}= (3x^2-a)^2$, where $a$ is a constant. Given that the curve has a stationary point at $(3,2)$, find the equation of the curve. I managed to get the equation $y=3x^3+3ax^2+xa^2$+c. I'm ...
0
votes
2answers
75 views

Proving that a function is Riemann Integrable

The usual definition to the Riemann integral is: for every $ε>0$, there exists $\delta$ such that if $P$ is a partition of $[a,b]$, and $\|P\|<\delta$, then $|S(f;P)-s|<\epsilon$. Then $f$ is ...
0
votes
1answer
47 views

If $f \le g$ and f, g are integrable, decreasing functions, then$\int_{x}^{\infty} f \le \int_{x}^{\infty} g$?

If $f \le g$ and $f, g$ are integrable, decreasing functions, then $\int_{x}^{\infty} f \le \int_{x}^{\infty} g$? Intuitively, I suppose it holds, but I have not found any such theorem in the ...
2
votes
3answers
43 views

Find expression in terms of x

Knowing that $$\frac{dy}{dx}= k\cdot x^{\frac{1}{3}}$$ and given that it passes through points $(1,4)$ and $(8,16)$, find an expression for the path in terms of $x$. I found out that $$y= \frac34 k ...
0
votes
1answer
30 views

Stein & Shakarchi, Complex Analysis, Ch.3 Ex.7

Suppose $f : \mathbb{D} \to \mathbb{C}$ is holomorphic, and $d = \sup_{z,w \in \mathbb{D}} |f(z) - f(w)|$. Show that $$ 2 |f'(0)| \leq d$$ This entire exercise is a complete mystery to me and I am ...