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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-2
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1answer
32 views

How to find the integral of $\int \frac{GMm}{r^2}\,dr$ [on hold]

I want to find the integral of: $$\int_R^\infty \frac{GMm}{r^2}\,dr$$
0
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2answers
28 views

How to use trigonometric substitution to compute this definite integral?

I have searched for a similar question on stack exchange but could not find one. The definite integral: $\large\int_0^1 \frac{x^4}{\sqrt{25-x^2}}$ I realize that I need to use $x = 5\sinθ$ in the ...
0
votes
2answers
39 views

What is happening to the '2' in this integral?

It is the indefinite integral: $\int \frac{1}{2x-6}$ I am trying to understand it and looking the last step goes from $\frac12 \log(2(x-3))$ to $\frac12 \log(x-3)$ Can someone explain to me why the ...
1
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0answers
20 views

Find the area (to three decimal places) bound by 2 equations

Find the area (to three decimal places) bounded by $f(x)=x^2e^x$ and $q(x)=4-x^2$ So far I've gotten $x^2(e^x+1)-4=0$ and the two $x$ values that make the equation $0$ are $1.027$ and $-1.86$ next I ...
0
votes
3answers
60 views

Evaluating $\int x^2 \sqrt{x^2-1} dx$

How do I evaluate the following indefinite integral? $$\int x^2 \sqrt{x^2-1} dx$$ Through integration of parts, I have obtained $$ \frac{x}{3}(x^2-1)^{3/2} - \frac{1}{3} \int (x^2-1)^{3/2} dx $$ ...
1
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2answers
45 views

Integration of $x/(x^2+1)$ from $-\infty$ to $\infty$

I am trying to find the area of this graph $\int_{-\infty}^\infty\frac{x}{x^2 + 1}$ The question first asks to use the u-substitution method to calculate the integral $incorrectly$ by evaluating ...
0
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0answers
34 views

A counterexample 2

Can we find a function $f:\mathbb{R}\to(0,\infty)$ which satisfies $$\limsup_{|x|\to + \infty}\frac{f(x+c)}{f(x)}<+\infty, \ \ \forall c\in \mathbb{R},(\text{limit in }+\infty\text{ and ...
1
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1answer
40 views

Why can we make this integral change of limits? Is it obvious?

When deriving the equation for the impulse-momentum theorem, the following occurs: $$\cdots=\int\limits_{t_1}^{t_2}\frac{d\vec p}{dt}dt = \int\limits_{\vec p_1}^{\vec p_2}d\vec p=\cdots$$ I know the ...
1
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2answers
40 views

Exponential Growth Differential Equation

A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is ...
3
votes
1answer
47 views

Improper integral of a cosine

I'm trying to follow some equations in an electrical engineering paper that I'm reading. I'll spare you the details, but at one point I come across: $$\lim_{ T \rightarrow \infty }\int_{-T/2}^{T/2} ...
0
votes
1answer
34 views

Symbolic Integration involving hypergeometric functions

What's the best way to symbolically evaluate this integral? $$\frac{1}{\hbar}\int_{-\infty}^\infty e^{iux/\hbar}\Psi^{*}_n(p-u/2)\Psi_n(p+u/2)\,du$$ where: $$\Psi_n(p)=\frac{1}{(1+\alpha ...
2
votes
1answer
72 views

Differential Equation $\frac{dy}{dt}$ = $y - t$

Given the differential equation $\dfrac{dy}{dt}$ = $y - t$ Is this equation separable? -> No it is impossible to separate this equation because we can't get $y$ alone with $dy$ and $-t$ alone with ...
1
vote
1answer
25 views

Need Help Understanding How To Integrate With An Implicit Variable

My calculus is really rusty (damn Mathematica/Matlab) and I was wondering if anyone could help me with an equation I am having trouble integrating. I have attached a snapshot of the paper I am trying ...
3
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0answers
52 views

Integral substitution paradox

Assume $f \in L^+(\mathbb{R})$ and $x>0$. Consider the integral $$ \int_0^\infty \frac{f\left(\frac{x}{y}\right)}{y} \: dy. $$ I am trying to make the substitution $u=x/y.$ I seem to get $$ ...
2
votes
2answers
52 views

When do evaluation and the integral sign “commute”?

This is a difficult question to put into words so it's much easier to write the math. Let $a$ and $b$ be given constants and $g(y) \equiv \int_a^b f(x,y) dx$. When is $g(c) = \int_a^b f(x,c) dx$? I ...
3
votes
1answer
26 views

an integral estimate from Stein's book, Singular Integral

I am reading the Stein's book Singular Integrals and Differentiability Properties of Functions. In the text (page 40), he states that $$ \int_{|x|\geq 2|y|}\Big|\frac{1}{|x-y|^n}-\frac{1}{|x|^n}\Big| ...
0
votes
1answer
35 views

A tricky integral with vanishing domain

I would love to have the following result, however I got no clue if it is even true! Let $B_n:=\{y:\varepsilon_n<|y|\leq\tilde{\varepsilon}_n\}$ for some sequences ...
1
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1answer
42 views

integrating square root of tanx [duplicate]

$\int \sqrt{\tan (x)}dx $ Let $\tan(x)=t^{2}$ then $dx$ will become $\frac{2t}{1+t^{4}}$ Hence $\int \sqrt{\tan (x)}dx =\int\frac{2t}{1+t^4} dt $ ...
2
votes
1answer
75 views

Examples of pairs of difficult integrals

I’m looking for pairs of difficult definite integrals that are linked algebraically on a certain field without known change of variable or integration by parts from one integral to the other. Here a ...
0
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0answers
39 views

Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$? [duplicate]

Given $f_n \rightarrow f$ a.e. Does $||f_n||_p \rightarrow ||f||_p$ imply $f_n\rightarrow f$ in $L^p$? Clearly this does not hold for $p = \infty$, since given functions with same hight, pointwise ...
0
votes
1answer
63 views

Evaluate $\int_0^{+\infty } \frac{\log(t)}{1+t^2} \, dt$ [duplicate]

How can we compute $$I=\int_0^{+\infty } \frac{\log(t)}{1+t^2} \, dt$$ Mathematica gives $I=0$.
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0answers
23 views

Integrating the logarithm of a function including a square root of a second degree polynomial

I have been trying for some time to calculate the following integral: $$\int \ln\left(k+\sqrt{ax^2+bx+c}\right)\ dx$$ where k, a, b and c are real numbers. I have tried several strategies, but without ...
0
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0answers
16 views

Finiteness of the lower integral implies finiteness a.e. of the function

I want to prove that if a function $f$ is $\mu$-measurable, $f\geq 0 $ $\mu$-a.e., then the integral of $f$ exists, that is its upper and lower integrals coincide. I've found the proof in Modern and ...
0
votes
2answers
22 views

Which is kernel similar gaussian kernel?

I must find a kernel that statisfies as follows: In the my reference paper, the author suggest gaussian kernel that is The purpose of that kernel is that it will take a weight for each points ...
1
vote
1answer
107 views

Prove $\int_{\mathbb{R^{+}}} \frac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\frac{\pi}{32} (3\pi-4)^2$

How do you arrive at the result $$I=\displaystyle\int_{\mathbb{R^{+}}} \dfrac{\sin^3 {(\pi x^2)} \cos {(4x^2)}}{x^5} dx=\dfrac{\pi}{32} (3\pi-4)^2\ ?$$ Wolfram Alpha agrees numerically. I tried ...
1
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0answers
57 views

How to do integral $\int_0^{\infty} e^{-x^2-ax^4}\ dx , \ \text{ for $a>0$}$

I was told by this OP, $$\int_{0}^{\infty} e^{\large-x^n} \,dx =\Gamma \left(\frac{n+1}{n}\right), \qquad\text{ for $n>1$}.$$ This is via the variable change $t=x^n$: $$\int_{0}^{\infty} ...
1
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0answers
15 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
0
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1answer
40 views

If $f$ is $+\infty$ on a set of positive measure and the integral exists in $[-\infty,+\infty]$, must the integral be $+\infty$?

Suppose $(X,\mathcal{M},\mu)$ is a measure space and $f$ a measurable function from $X$ to $[-\infty,+\infty]$. Suppose that $$\int_{X}f\ d\mu$$ exists in $[-\infty,+\infty]$, and that $X$ contains a ...
1
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0answers
54 views

How do you calculate the Riemann zeta function of a complex number given the complex contour integral?

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
1
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1answer
52 views

An introduction for integral tricks.

I wonder if there's a good book or internet page introducing integral tricks? For example integration by parts, and Feynman's trick. I'm not looking for an exercise book such as "Problems in ...
1
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4answers
24 views

Find depth of a half-filled parabolic cross-section

Given a cross-section of an object that is parabolic in shape, how do you find the depth of the object when it is "half full". A full example given in an exam: A long trough whose cross-section ...
0
votes
2answers
105 views

Improper integral of $\frac{\ln x}x$

Find $$\int_e^{\infty}\frac{\ln x}{x}\ dx$$ $A.\ \dfrac12$ $B.\ \dfrac{e^2}{2}$ $C.\ \dfrac{\ln(2e)}{2}$ $D.$ DNE (Does not exist) I tried doing this and this is where I've gone so far: $$\lim ...
2
votes
3answers
112 views

Separable differentiable equations

Which of the following is a solution to the separable differentiable equation: $$\frac{dy}{dx}=\frac{xy}{\ln y }$$ $A.\ \displaystyle e^{|x|}$ $B.\ \displaystyle e^{\sqrt{\frac{x^2}2}}$ $C.\ ...
0
votes
0answers
33 views

What is this called specifically?

Imagine you take a radius from the center of the shape, you add up all of the lines as it rotates 360 degrees. The radius is measured from its point of rotation, like (0,0) in Cartesian coordinates,to ...
4
votes
1answer
166 views

Can we express the following in a closed form? [duplicate]

I want to evaluate the integral: $$I=\int_{0}^{\pi/2}\ln \left ( \frac{(1+\sin x)^{1+\cos x}}{1+\cos x} \right )\,dx$$ Well, the sub $u=\pi/2-x$ does not give me any result. In fact it makes the ...
3
votes
0answers
39 views

Why is the value assigned to a gauge integral well defined (unique)?

Why is the value assigned to a gauge integral well defined (unique)? If we would have given a net (so an underlying order that happens to be directed), then the limit would be unique given a ...
0
votes
1answer
34 views

$f(r) \leq \int_r^{r+1} f(t)dt$

Suppose $f:[0,\infty)\to [0,\infty)$ is continuous (uniformly, if you want) and that $\int_0^{\infty} f(t)~\mathrm{d}t < \infty$. Is the following true? $$ f(r) \leq \int_r^{r+1} f(t)~\mathrm{d}t ...
3
votes
1answer
21 views

What is the value of $a$ so that this condition holds?

Let $f(x) \colon= x-x^2$, $g(x) \colon= ax$. Determine the value of $a$ so that the region above the graph of $g$ and below the graph of $f$ has area equal to $9/2$. Here $f(x) - g(x) = (1-a)x - x^2 ...
1
vote
2answers
54 views

How are these two integrals related?

How to express the integral $$\int_{-2}^{2} (x-3) \sqrt{4-x^2} \ dx $$ in terms of the integral $$ \int_{-1}^{1} \sqrt{1-x^2} \ dx?$$ I know that the latter integral is equal to $\pi / 2$. We can't ...
0
votes
0answers
35 views

Contour integration with merged pole/branch-cut type behavior?

I have the expression $$f(z)=\frac{-i}{\sqrt{z^2-a^2}},$$ where $a$ is a purely real number and $z$ is a complex variable. Numerical plotting gives the following. This leads me to the following ...
3
votes
1answer
56 views

Integral of products of cosines

Given $m+1$ integers $\alpha_0,\ldots,\alpha_m\geq 1$, I was trying to get a nice closed formula for the integral $$ \int_0^\pi\cos(\alpha_1\theta)\cdots\cos(\alpha_m\theta)d\theta. $$ More precisely, ...
2
votes
2answers
27 views

Double integral where limits are the first quadrant

Evaluate the integral $$\iint\limits_D \frac{1}{(x+y+1)^3} \, dA$$ where $D$ is the first quadrant. In this case, what would the limits of integration be? I'm having trouble moving to polar ...
1
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2answers
35 views

Asymptotic conditional distribution of normal variable

$X$ is a normal variable $\mathcal{N}(0,1)$, $Y$ is a normal variable $\mathcal{N}(n,n-1)$, independent of $X$. I want to prove that the distribution of $X$ conditionally on $X > Y$ is ...
0
votes
0answers
26 views

$ L^{2} $ convergence should imply convergence in infinity norm

My situation is the following: suppose we have a Lie group $ G=G_{1} \times G_{2} $ and let $ X = \Gamma \backslash G $ a homogeneous space arising from a lattice, i.e. we have a $ G $ invariant ...
-3
votes
1answer
40 views

Calculus use of integral [on hold]

Assume that the price of a product is at a constant value of $\$100$ per unit or the marginal function is $MR=f(x)=100,$ where $x$ equals the number of units sold $a)\ $ What is the total revenue ...
1
vote
3answers
107 views

Evaluate $\int_0^\infty\frac{dl}{(r^2+l^2)^{\frac32}}$

How to evaluate the following integral $$\int_0^\infty\frac{dl}{(r^2+l^2)^{\large\frac32}}$$ The solution is supposed to look like this, unfortunately I can't derive it. $$ ...
2
votes
3answers
173 views

Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
3
votes
2answers
50 views

How to determine the point at a set length along a given function (parabola)?

Given a specific function, a parabola in this instance, I can calculate the length of a segment using integrals to sum infinite right angled triangles hypotenuse lengths. My question is, can I reverse ...
-2
votes
0answers
48 views

What does this complex contour integral mean? [on hold]

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
1
vote
1answer
30 views

Work done by a force field line integrals

Find the work done by the force field $F(x, y) = \langle 2x \sin(y), 2y \rangle$ on a particle that moves along the parabola $y = x^2$ from $(-1, 1)$ to $(2, 4)$. So to use line integrals to solve ...