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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Sphere growing rapidly. Determine radius given volume.

So I have this balloon right? I am blowing air into it and it's volume is increasing at 4cm^3/s. I want to know what the rate of change will be when the balloon's radius is equal to 10cm (assuming ...
2
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0answers
32 views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
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1answer
29 views

Volume of the solid with given base, whose sections with the planes orthogonal to $y = 0$ are rectangles of height $4$

Please help me to solve the following problem: Determine the volume of the solid having as base the portion of cartesian plane limited by $y = 0$ and by $y = x^{3}$ in the closed interval ...
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1answer
29 views

Question about the Fundamental Theorem of Calculus

So I have studied the FOTC, but not really sure of what I read so this question is just to help me learn the FOTC and understand how to do problems like it. $$ if $$ $$F(x)=\int_0^x\sqrt{sin^3(t)}dt$$ ...
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0answers
38 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
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2answers
41 views

How can you explain implicit differentiation?

So I am taking calculus 1 online from a local college (bad idea, but the only thing that fit my schedule). The professor used the notation $f'(x) =$ for EVERY function up until two weeks ago. All of ...
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1answer
219 views

Can all integration be thought of as projections?

For example, the integral of the function f(x) could be thought of the projection of f on the function g, where g is identically 1. Following this logic, can we think of the multiplication of f and ...
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24 views

How to evaluate the following integral? $∫_{-β}^{2π-β}\exp⁡(ix\cos(φ-β))dφ.$

I'm trying to calculate the following integral: $$∫_{-β}^{2π-β}\exp⁡(ix\cos(φ-β))dφ.$$ I tried by parts with no success and also by writing $\exp (ix)$ in terms of $\sin$ and $\cos$, with no ...
2
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1answer
54 views

Evaluating $\int^{\frac{\pi}{2}}_0 \sin^n x ~\mathrm{d}x$

I'm trying to find the general formula for the following: $$I_n = \int^{\frac{\pi}{2}}_0 \sin^n x ~\mathrm{d}x$$ I remember doing it a while back but for the life of me, I can't remember right now. ...
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0answers
58 views

Help on the Integration of $\int_0^{\infty} e^{-bx}\sin ax^2 \, \mathrm{d}x$.

I have had the misfortune of coming across the following integral, for real $b$ and $a > 0$: $$\int\limits_{0}^{\infty} e^{-bx} \sin\left(ax^{2}\right) \, \mathrm{d}x.\tag{1}$$ Naturally, I ...
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9 views

Heuristic Algorithm for integrating algebraic functions

Is there any heuristic "algorithm" or a good technique for integrating algebraic functions? The general algebraic case was solved by Trager and Davenport. But their algorithms are rather complicated ...
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2answers
40 views

Help on an integration by substitution

In a proof to show that $\int_{0}^{1} f \left(\left\{1/x\right\}\right) \frac{ \mathrm{d}x}{1-x}=\int_{0}^{1} f(v) \frac{ \mathrm{d}v}{v}$, i found this line : ...
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0answers
21 views

what are and why are sine and cosine modulated integrals used?

I have found the definition of the following formulas in a paper, where they are called sine and cosine modulated integrals. $y$ is a signal with a strong periodic component of frequency $N\Omega$ ...
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0answers
12 views

Simplification of an integral comprising of vector-variables

How can I evaluate the simplify the integral $\int \rho (\bf{r^{\prime}})\, \delta (\frac{\sigma}{2}-r) d \bf{r^{\prime}}$ where $\delta$ is the dirac-delta function given that $\rho$ is constant ...
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3answers
78 views

Integral of $\int_{0}^{y} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x$

Does the following integral have a closed form solution? $$ \int_{0}^{y} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x $$ Or is there an approximation which works for large $\alpha$?
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1answer
53 views

Integration of powers: nested $dx$?

How do we solve the likes of the following expression: $$ \int_0^2 \frac{x\,dx}{\sqrt{1 + 2x^2}}\,dx $$ I'm bothered by the nested $dx$ in the numerator. How is this solved using the general power ...
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1answer
46 views

Is it true that $\int_{-C} f(x, y)ds = -\int_C f(x, y) ds$ [on hold]

I think it is more of a convention question, right ? $$\int_{-C} f(x,y)ds = -\int_C f(x,y) ds$$
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0answers
66 views

Connection between integral expression and the factorial of infinity [on hold]

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also ...
3
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2answers
146 views

Integral of exponential with $x(1-x)$ term

Does the following integral have a closed form solution? $$ \int_{0}^{y} \exp\left(\,\sqrt{\,x(1-x)\,}\,\right)\,{\rm d}x $$ Or must I settle with an approximation? Edit: Actual form of integral ...
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1answer
88 views

How to find the derivative of the function $ \int_{x}^{x^2}\frac{t}{\ln(t)}dt$? [on hold]

The problem is to find $\displaystyle\frac{d}{dx}\int_{x}^{x^2}\frac{t}{\ln(t)}\,dt$ I could do this if I had the first clue on how to integrate $\dfrac{t}{\ln(t)}$ but even wolframalpha is giving ...
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2answers
27 views

Finding the Value of K in an Integral Function

Given the function $$f(x)\begin{cases} -2(x+1), & \text{x $\le0$} \\ k(1-x^2), & \text{x $\gt0$} \\ \end{cases}$$ Find the value of k for $$\int_{-1}^1f(x)dx=1$$ Wasn't really sure how to ...
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4answers
126 views

How would I go about evaluating $\int \frac{x}{(9-8x^2)^3}dx$?

So I have homework on webAssign (a site used by my college), and I am not understanding the logic as to why I am taking the steps into solving the integral it is telling me to take. So I'll list the ...
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0answers
12 views

Non Borel Spaces: Gauge Integral

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
2
votes
3answers
84 views

Evaluate $\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} $

$$\int_{1}^{\infty} \frac{\ln{(2x-1)}}{x^2} dx$$ My approach is to calc $$\int_{1}^{X} \frac{\ln{(2x-1)}}{x^2} dx$$ and then take the limit for the answer when $X \rightarrow \infty$ However, I must ...
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3answers
33 views

Integral of a function with two parts (piecewise defined)

The function has 2 parts: $$f(x) = \begin{cases} -\sin x & x \le 0 \\ 2x & x > 0\end{cases}$$ I need to calculate the integral between $-\pi$ and $2$. So is the answer is an integral ...
2
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2answers
25 views

Integration with two unknowns

I'm completely stumped with this one, I'm not sure how I should do this. The equation of a parabola is $y=-3x(x-2)$. It intersects the $x$-axis at $0$ and $2$. Given that the area of this parabola ...
5
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4answers
218 views

Integrate a division of polynomials

Hi I have the following integral: $$\int \frac{2x}{x^2+6x+3}\, dx$$ I made some changes like: $$\int \dfrac{2x+6-6}{x^2+6x+3}\, dx$$ then I have: $$\int \dfrac{2x+6}{x^2+6x+3}\, dx ...
4
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4answers
739 views

Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
2
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2answers
54 views

How do you prove this integral involving the Glaisher–Kinkelin constant

According to wikipedia on the page Glaisher–Kinkelin constant $$\int_0^{1/2} \ln\Gamma(x) dx=\frac32\ln \text{A}+\frac5{24}\ln 2+\frac14\ln\pi$$ I got interested in how you possibly could prove ...
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2answers
18 views

Average Value - Graphs

long method: Determine an equation for each and solve using average value formula alternative methods? How could you prove the average value to be C over an interval [a,b] if you are given a ...
0
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1answer
8 views

Left & Right Area Approximation Using Y-Axis - Method Alternatives

Is there a simpler way of solving this then calculating x1(h)+x2(h)+x3(h)+x4(h) by using the given y values (in this case h, the height is one, because the length of each rectangle is one) ...
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1answer
9 views

Related Rates of Change - Cylinder Question

A cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me correct ...
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0answers
26 views

Darboux integrable, $f$ continuous at x where g(x)=G(x) [duplicate]

$f:[a_1,b_1]x[a_2,b_2]\rightarrow \mathbb{R}$ that is Riemann integrable, and let $g(x),G(x)$ functions with property $g(x)\leq f(x) \leq G(x)$, g=G a.e.! G(x), g(x) are obtain from proof Riemann int ...
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0answers
25 views

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta$, where a and b are real numbers. If we consider ...
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1answer
18 views
1
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1answer
31 views

Inner product of functions as integration

I am trying to teach my self some linear algebra in preparation for a module in machine learning. I am using Gilbert Strang's text Introduction to Linear Algebra and am having some difficulties. My ...
1
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1answer
24 views

Integration of a matrix by MATLAB

How do I integrate a matrix in MATLAB: A=[1,2;3,4]; B=[2*t;t^2]; i.e, how to compute: integral{expm(A*s)*B(s)}ds between ...
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1answer
36 views

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

I want to find the integral of: $$\int_R^\infty \frac{GMm}{r^2}\,dr$$
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2answers
38 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
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2answers
42 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 ...
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0answers
25 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 ...
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3answers
81 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 $$ \frac{x}{3}(x^2-1)^{3/2} - \frac{1}{3} \int (x^2-1)^{3/2} dx $$ ...
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4answers
69 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 ...
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0answers
39 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 ...
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2answers
45 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
2answers
54 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
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1answer
44 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
76 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
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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 ...