All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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4
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4answers
110 views
4
votes
4answers
104 views

How find this integral $I=\int_{0}^{\frac{\pi}{2}}(\ln{(1+\tan^4{x})})^2\frac{2\cos^2{x}}{2-(\sin{(2x)})^2}dx$

Find the value: $$I=\int_{0}^{\frac{\pi}{2}}(\ln{(1+\tan^4{x})})^2\dfrac{2\cos^2{x}}{2-(\sin{(2x)})^2}dx$$ I use computer have this reslut ...
2
votes
1answer
37 views

Computing a contour integral over curve not centered at origin

Consider the integral $$ \int_C \frac{1}{z} \, dz $$ where $C$ is the circle of radius $R$ centered at the point $z_0 \in \mathbb{C}$. We parametrize the curve by $z(\theta) = z_0 + Re^{i\theta}$ ...
2
votes
1answer
32 views

Does the limit $\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$ always exist?

Let $f$ be a Lebesgue integable function. Does the limit $$\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$$ always exist?
1
vote
0answers
15 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
0
votes
2answers
52 views

How can the double intergal expression be reduced to the single intergal expression

Consider the following expression where $x(s)$ and $y(s)$ are continuous as is necessary on the closed interval [a,b]. (This is a functional analysis question -- see below for details.) $$x(s) = ...
20
votes
4answers
479 views

A closed form for $\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx$

I need to a evaluate the following integral $$I=\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx.$$ Both Mathematica and Maple failed to evaluate it in a closed form, and lookups of the ...
0
votes
1answer
28 views

Where is the error in this parameterization?

The problem is thus: find $$\int_c (x+2y)\mathrm{d}x+x^2\mathrm{d}y \space \mathrm{where \space C \space consists \space of \space \space line \space segments \space} (0,0)\space \mathrm{to} \space ...
-1
votes
1answer
50 views

If $f$ is in $R[a,b]$, show that [on hold]

If $f$ is in $R[a,b]$, show that $$\int_a^b f =\lim_{c\to a^+} \int_c^b f$$ the hint is to show $$\left|\int_a^b f - \int_c^b f \right| = \left|\int_a^c f \right| \le \int_a^c |f|$$
0
votes
2answers
67 views

Show that the given function is R-Integrable

Show that the function $f(x)=[x^2]+|x-1|$ is Riemann Integrable on $[0,2]$ where $[\alpha]$ denotes the greatest integer less than or equal to $\alpha$ Defining the function over the interval i get ...
4
votes
1answer
106 views

What is the error in this line integral?

C is the arc of the curve $y=\sqrt{x}$ from $(1,1)$ to $(4,2)$. Find $$\int_cx^2y^3-\sqrt{x}\space\mathrm{d}y$$Looks simple enough. I take $x=t$ and $y=\sqrt{t}$. This leaves $$\int_1^2[t^2\cdot ...
0
votes
1answer
113 views

Normalization of a two-dimensional kernel function

I've got three two-dimensional kernel functions which look like this $$ k(r,h) = n \cdot \begin{cases} \ldots & 0 \le r \le h \\ 0 & otherwise \end{cases} $$ With ...
0
votes
1answer
75 views

Need help with a integral

I was evaluating $$ \int_{0}^{^\pi/_2}x\ln\left(\vphantom{\large A}\cos\left(x\right)\right)\,{\rm d}x $$ I like to try with the fourier series $$ \int_{0}^{^\pi/_2} \left[\,\,\sum_{k = 1}^{\infty} ...
1
vote
4answers
39 views

Using integrating factor

I have the following differential equation $$\frac{dN(t)}{dt} = A - \mu N(t)$$ I understand that I will need to use an integrating factor but am not sure how to proceed. I think I should use ...
2
votes
2answers
56 views

Integration by Tables problem

$$\int \frac {dx} {x(x^8-256)}$$ I am supposed to use the formula $$\int \frac {dx} {x(ax+b)} = \frac1b\ln\left|\frac x {ax+b}\right|+C $$ to find the integral. I don't know how to start. Help is ...
3
votes
2answers
53 views

Evaluate integral by completing the square and doing trigonometric substitution

$\int \frac{1}{(x-2)\sqrt{x^{2}-4x+3}} dx$ is my problem Complete the square $\int \frac{1}{(x-2)\sqrt{(x-2)^{2}-1}} dx$ I know I'm probably supposed to use $ \frac{d}{dx}\operatorname{arcsec}(u) = ...
1
vote
1answer
39 views

inequality property of integrals

Suppose $f(x), g(x), h(x)$ are non-negative real valued functions on $[a, b],\;\;(0\leq a<b)$ and $p$ is any positive real number and $f(x)^p \leq g(x)^p+ h(x)^p.$ Is it true that ...
2
votes
1answer
704 views

Proving Abel-Dirichlet's test for convergence of improper integrals using Integration by parts

I'm struggling with the following calculus question. Let there be two functions $f,g : [a, \infty) \to \mathbb R$ such that: $g$ is monotonic, differentiable and has a limit at zero $f$ is ...
11
votes
3answers
343 views

Finding the maximum value of $\displaystyle \int_{0}^{1}e^x\log f(x)dx$ when $\displaystyle \int_{0}^{1}f(x)dx=1$

Suppose that $f(x)\ (0\le x\le 1)$ is continuous and strictly positive and satisfies $$\int_{0}^{1}f(x)dx=1.$$ Then, can we find the maximum and the minimum value of the following? If yes, then how? ...
2
votes
1answer
56 views

Integral involving Gamma Function

I am solving the following integral: $$ \int_{-1}^{K} u^B e^{-u} du $$ The solution of the integral is a lower incomplete Gamma Function if -1 is replaced with 0. Can anybody help me in solving the ...
6
votes
4answers
374 views

A generalized integral need help

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
1
vote
2answers
120 views

Evaluate $\int_0^1 \sqrt{2x-1} - \sqrt{x}$ $dx$

I'm trying to calculate the area between the curves $y = \sqrt{x}$ and $y= \sqrt{2x-1}$ Here's the graph: I've already tried calculating the area with respect to $y$, i.e. $\int_0^1 ...
1
vote
0answers
17 views

Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
1
vote
0answers
22 views

Proof for Scheffe's Lemma and General Dominated Convergence theorem

While reading this question here about the proof for Scheffe's Lemma, I was confused since someone said the proof in the question was not correct. I thought the argument was fine, and the author only ...
1
vote
2answers
67 views

Triple Integral exercise

Calculate $\int\int\int_Dz\;dxdydz$ if $D$ is the region inside $z=0,z=\sqrt{x^2+y^2}$ and $x^2+y^2=1$. I would like to know if the answer I got is right. This is what I did: $(1)$ Change to ...
2
votes
0answers
65 views

How to evaluate the integral $\int e^{ipx}e^{ipx} d^{3}x = 0$

I am embarrassed to ask this question. But I came across the following in a physics book: $$\int e^{ipx}e^{ipx} d^{3}x = 0$$ $d^{3}x = dydydz$, as @Semiclassical shows below. This came up in the ...
1
vote
2answers
38 views

$\dfrac{\partial}{\partial x}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$

I'm trying to prove the following, interesting, relation: $\dfrac{d}{dx}\left(\int_{g(x)}^{h(x)}f(y)\, dy \right)= f(h(x))h'(x)-f(g(x))g'(x)$ I tried to integrate by parts the RHS, but i don't ...
4
votes
1answer
39 views

Double Integral transformation to Polar coordinates

Here's the question from an exam that I couldn't solve: If $\int_1^2 \int_0^x \frac{1}{(x^2+y^2)^\frac{3}{2}} ~\mathrm{dy} ~\mathrm{dx}$ transforms to $\int_0^a \int_b^c \frac{1}{r^2} ...
2
votes
1answer
81 views

Changing a double integral into polar coordinates

I have the double integral $\int^a_0\int^{\sqrt{a^2-x^2}}_0 e^{-(x^2+y^2+a^2)} dydx$ And I am asked to evaluate this by changing to polar coordinates. I know the transformations are, $x=r ...
2
votes
3answers
129 views

Integration - finding an explicit formula

The question in my textbook asks: If $f$ is a continuous function such that $$\int\limits_0^x{f(t)dt}=xe^{2x}+\int\limits_0^x{e^{-t}f(t)dt}$$ for all $x$, find an explicit formula for $f(x)$. My ...
1
vote
2answers
43 views

Doubt in integral substitution

I am not able to figure out what substitution to use in the following integral $$ \int \frac{(x-1)e^x}{(x+1)^3}dx $$ Any help would be appreciated.
2
votes
2answers
49 views

Examples of interesting integrable functions with at least 2 fixed points and an explicit inverse

What are some interesting functions I can use to demonstrate this integration trick: $$\int_a^b [f(x)+f^{-1}(x)]=b^2-a^2$$ I would like to know of some interesting functions where this trick is not ...
1
vote
1answer
95 views

Trying to solve $\int{-2\exp{\left(z\cos^2 \theta \frac{\left(a^2 - 1\right)}{2a^2}\right)}}d\theta$

I am trying to solve this integral which has come up as part of some other work, but it is proving to be much harder than I had originally thought. For $0 < |a| \le 1$ being some constant, I am ...
5
votes
2answers
99 views

Is $\int^x \cos \frac1t$ differentiable at zero?

From Spivak's Calculus, 4th ed., exc 14-20: Let $$f(x) = \begin{cases} \cos \frac1x, & x\neq 0\\ 0, &x=0. \end{cases}$$ Is the function $\int_0^xf$ differentiable at zero? I'm having ...
2
votes
1answer
80 views

Inconventional Integral inequality

$$\int_a^bw(x)|f(x)||g(x)|\;dx \le \left(\int_a^bw(x)\;dx\right) \max_{a\le x\le b}|f(x)|\cdot \max_{a\le x\le b}|g(x)|$$ I don't really understand this integral inequality. How do I go about ...
2
votes
3answers
109 views

Evaluate $\int x e^{\sqrt{x}} \, dx$

$$\int_0^1 xe^{\sqrt{x}} dx = ? $$ All I can think of is the integration by parts rule, where $ u = x $ and $ dv= e^{\sqrt(x)} $ $ \Rightarrow du = 1$ and $ v= e^{\sqrt(x)} $ The answer I get is ...
1
vote
2answers
98 views

How does the integral $\int_{D_C} e^{ia z}P(z)/Q/(z)\,\mathrm{d}z$ blow up.

In my book I have a theorem that goes something like the following Let $P(x)$ be $Q(x)$ polynomials such that $\deg(Q) \geq \deg(P) + 2$. Then \begin{align*} \int_{-\infty}^{\infty} ...
12
votes
1answer
143 views

$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$ [duplicate]

$$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$$ I tried Integration by parts but failed. Wolfram alpha gives answer in decimal points which are same as of $2\pi$. Any hints or suggestions will be helpful.
0
votes
2answers
67 views

Integrate $\int \sin^4x \cos^2x dx$

Integrate $$\int \sin^4x \cos^2x dx$$ Now, there's few solutions to this problem already on the internet. For example on yahoo: https://answers.yahoo.com/question/index?qid=20090204203206AAbjUfM and ...
3
votes
5answers
192 views

Calculating the area

For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them; Attempt to solve: Limits of the integral are $1$ and $-3$, so I took ...
2
votes
1answer
43 views

Problem with volume integral of a scalar function

I have difficulties in integrating this scalar function over the assigned volume. Let $D=\{(x,y,z):(x-2z)^2+(y-x)^2+(x+z)^2\le4,\,0\le x+y+z\le1\}$ Calculate $\int_D z\,dxdydz$
1
vote
3answers
88 views

Evaluate $\int \frac{1}{(2x+1)\sqrt {x^2+7}}dx$

How to do this indefinite integral (anti-derivative)? $$I=\displaystyle\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}dx$$ I tried doing some substitutions ($x^2+7=t^2$, $2x+1=t$, etc.) but it didn't work out.
4
votes
0answers
52 views

Clarification of Contour Integration [duplicate]

I apologise if this seems like an elementary and silly question, but I am confused about the integral $$I=\int^{\infty}_{-\infty}\frac{\cos{x}}{1+x^2}dx=\frac{\pi}{e}$$ If I consider a semicircular ...
1
vote
2answers
40 views

Euler's method for first three approximations?

I have tried variations of the problem for an hour at least and cannot get around to sloving this one. Thank you for input!
2
votes
1answer
87 views

Improper integral $\int_{0}^{\pi} \frac{x}{\sin x} dx$

Find out whether or not the following integral exists $$\int_{0}^{\pi} \frac{x}{\sin x} dx.$$ I'm pretty sure this integral doesn't exist but I can't seem to find a good way to prove this. It ...
2
votes
3answers
273 views

Distribution of stochastic integral

Assume that $\mathrm{d}S = \sigma \, \mathrm{d}W$ with initial level $S(0)$ and where $\mathrm{d}W$ is usual Brownian motion. Now $$A(T) = \frac{1}{T} \int_0^T S(t) \, \mathrm{d}t.$$ ...
7
votes
2answers
205 views

Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of ...
2
votes
1answer
32 views

Product measure and integrals of simple functions

Let $(\Omega_1 , \mathcal{X}, \mu)$ and $(\Omega_2 , \mathcal{Y}, \nu)$ be two $\sigma$-finite measure spaces, and let $\mu \times \nu$ be product measure on the $\sigma$-algebra $\mathcal{X} \times ...
0
votes
0answers
35 views

The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
2
votes
1answer
47 views

Solving indefinite integrals gives multiple answers. Are all those answers correct?

While solving problems on indefinite integrals many a times I get answers which are different from those given in my text book's answer keys page. I then verify my solution steps to ensure that even ...