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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1answer
23 views

Norm square of an integral

Is it allowable to do like this: $$ \Psi(x,t) = \int_{-\infty}^{+\infty} e^{k^2/a}*e^{ikx}*e^{-ik^2t} $$ $$ |\Psi(x,t)|^2 = |\int_{-\infty}^{+\infty} e^{k^2/a}*e^{-ikx}*e^{-ik^2t}dk|^2 $$ $$ ...
-4
votes
2answers
24 views

Problem 3: Find $\int xe^xdx$ by using appropriate integration techniques. [5 marks]

Problem 3: Find $$\int xe^xdx$$ by using appropriate integration techniques. [5 marks]
3
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1answer
632 views

Approximating integrals with step functions

For $f \colon [1,2] \to \mathbb{R}$ , $f(x) = 1/x$, Choose a sequence of step functions $\phi_n$ approximating $f$ with partition $P_n := [r/n : n < r < 2n]$ to show that $ 1/(n+1) + \cdots + ...
1
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1answer
48 views

List of general Integral proposed for authors that produce the simplest of proof

On this page we are intended on proposing integrals with closed form. We are looking for; for a simple solution by any of the authors. Here we are not asking for, of any of the authors to find us the ...
0
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1answer
27 views

How do I get this integral? [duplicate]

This is from a list of problems from a local university's Integration Bee. I have no idea how to do it, but I thought maybe someone here can explain it to me. $$\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} ...
0
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0answers
13 views

Banach Theorem on Metric Space for Integral Equations

My instructor said that Banach doesn't apply in this case: f(x) = sin(x) + $\int_0^x$$f^2$(z)dz f(0) = 0; f'(0) = 1 > 0 f'(x) = cos(x) +$f(x)^2$, which is positive on (0,$\pi$) so f is positive ...
2
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1answer
39 views

Prove that $\lim_{n \to \infty}\left(\int_{a}^{b}|f(x)|^ndx\right)^{\frac{1}{n}} = \sup_{x \in [a,b]} |f(x)|$

$f:[a,b] \rightarrow \mathbb{R}$ is continuous.Prove that $$\lim_{n \to \infty}\left(\int_{a}^{b}|f(x)|^n dx\right)^{\frac{1}{n}} = \sup_{x \in [a,b]} |f(x)|$$ I was thinking of Holder's inequality ...
0
votes
3answers
73 views

Show $\int_0^1 \frac{\log(1-x)}{x}=-\frac{\pi^2}{6}$

It's claimed that $$\int_0^1 \frac{\log(1-x)}{x}=-\frac{\pi^2}{6}$$ by first expanding $\frac{\log(1-x)}{x}$ into a power series and then doing term-by-term integration. I want to justify this by ...
0
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2answers
19 views

Change integration limits, multivariable calculus.

Good night, i have a serious problem changing the integration limits, i read two books but i don't understand, i put an example... $\int_{0}^{1}\int_{0}^{1-x}\sqrt{x+y}\left(y-2x\right)^{2}dydx$ I ...
2
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1answer
372 views

Confused about notation and derivatives inside integrals

EDIT: To make what I am asking more clear. I've simplified it and have a more direct question. Let's say I am writing out an expression, and I want to write: $$\int_0^xF'(y)\,dy$$ However, for ...
0
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0answers
18 views

Intergration $\dfrac{1}{x}\exp\left[i(Ax^2+Bx+C)\right]$

I need to calculate the integral: $$\int^{\infty}_{0}\dfrac{1}{x}\exp\left[i(Ax^2+Bx+C)\right]$$ I guess complex analysis is suitable for this integral, but I still have no ideas which kinds of trick ...
4
votes
2answers
54 views

Areas under the graphs of $\frac{1}{x}$ and $\frac{1}{x^2}$ from $1$ to $\infty$

A simple evaluation of the definite integral tells us that the area under the graph of $[\frac{1}{x}]^2$ from $1$ to $\infty$ is finite whereas that of $\frac{1}{x}$ for the same limits is infinite. ...
3
votes
1answer
65 views

Brostein Integral 21.42

Good morning. I came across the following integral in some field theory calculation: $\int_0^\pi dx\,\log\left(a^2+b^2-2ab\cos x\right)=2\pi\log\left(\max\lbrace a,b\rbrace\right)$ for ...
2
votes
1answer
495 views

Find the complete integral of $(p+q)(px+qy)=1$.

I am stuck on the following problem that says: Find the complete integral of $(p+q)(px+qy)=1$,where $p={ \partial z \over \partial x},q={ \partial z \over \partial y}$. My Attempt: The ...
0
votes
1answer
940 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
-2
votes
1answer
38 views

Double obstructing wall problem, what is the optimal walk path and length?

Every day, you walk from point A to point B which are exactly $2$ miles apart straight line distance, however, each day, there is a $50$% chance of there being an obstructing wall perpendicular to the ...
1
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4answers
90 views

$\int \frac{\cos(x)}{(1+\cos(x))^3} \, dx$?

I'm having a bit of a trouble seeing how to factorize the result of substituting $t=\tan(x/2)$, $\cos(x)=\frac{1-t^2}{1+t^2}$ and $dx=\frac{2}{1+t^2} \, dt$ into $$\int \frac{\cos(x)}{(1+\cos(x))^3} ...
0
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1answer
24 views

Finding the domain of the following integral in polar coordinates

Question: Convert the following integral into polar coordinates and solve $$\int_0^\frac{\sqrt{2}}{2}\int_x^\sqrt{1-x^2}xy \ dy\,dx$$ My attempt: I managed to get this: ...
1
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1answer
30 views

Calculate $\int_D \rvert x-y^2 \rvert dx \ dy $

$$\int_D \rvert x-y^2 \rvert dx \ dy $$ $D$ is the shape that is delimited from the lines: $$ y=x \\ y=0 \\ x=1 \\$$ $$D=\{ (x,y) \in \mathbb{R}^2: 0 \le x \le 1 \ , \ 0 \le y \le x \}$$ ...
0
votes
2answers
50 views

What is y'' if $\sin y = y + 5x$?

I got $ 5\sin y / (\cos y - 1)^2$ as my answer, but the correct answer was given as $25\sin y / (\cos y - 1)^3$. My thought process: Derive the original equation to get $y'\cos y = y' +5$ $$y'(\cos ...
0
votes
2answers
58 views

Antiderivative for $\sin(t^2)/2$?

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
votes
1answer
24 views

Questions on multidimensional integrals

$$ f(x,y) = \begin{cases} x^2(1 − y)^2 & 0 ≤ x ≤ y ≤ 1 \\ x + y & 0 ≤ y < x ≤ 1 \end{cases} $$ Argue that $f$ is continuous at all points in $[0, 1] × [0, 1]$ that are not in the closed ...
1
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1answer
18 views

Can gauss quadrature integrate this function exactly $f(x) = \frac{2x}{\sqrt{x^3 + 2x + 5}}$?

Suppose I had a function $$f(x) = \frac{2x}{\sqrt{x^3 + 2x + 5}},$$ that I wanted to integrate on the interval $[\pi, 2\pi]$. Can Gauss quadrature of order $2$ (ie. with two points ...
3
votes
0answers
19 views

Extending the Riemann integral to $(-\infty, \infty)$

It's natural to define the integral $$\int_{0}^\infty f(x) dx := \lim_{M\to\infty} \int_{0}^M f(x) dx~~~~~~ ~~~(*)$$ But it's not obvious how we should define the integral $$\int_{-\infty}^\infty ...
2
votes
1answer
90 views

How can we see that $ \sum_{n=0}^{\infty}\frac{2^n(1-n)^3}{(n+1)(2n+1){2n \choose n}}=(\pi-1)(\pi-3) $?

I wonder will it help me so prove it if I was to decompose it into partial fractions? Mathematica approves of the identity; it is converges. can anyone help me to prove it? $$ ...
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votes
2answers
76 views

Easiest way to solve this integral [on hold]

I was solving this problem from a calculus textbook and I got stuck at this particular problem. I tried to put it into Integral Calculator after I was unable to solve it, but now I wonder if there is ...
2
votes
2answers
25 views

Is there a solution to this differential equation?

I am trying to find a function $y(x)$ that is a solution to $$ \left(a_3 x^3+a_1 x\right) y''(x)-\left(3 a_3 x^2+2 a_1\right) y'(x)+3 a_3\, x \,y(x)=a_0 x^4+a_2 $$ I tried using mathematica but it ...
1
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3answers
56 views

Prove that $\int_{0}^{\frac{\pi}{2}}{\frac{\sin(2n+1)x}{\sin(x)}dx}=\frac{\pi}{2}$ for $n\ge0$

Prove that $$\int_{0}^{\frac{\pi}{2}}{\frac{\sin(2n+1)x}{\sin(x)}dx}=\frac{\pi}{2}$$ for $n\ge0$ I am not able to proceed with the integral. For the case $k+1$ please guide me through the problem. ...
2
votes
1answer
91 views

What's the integral of $\frac{1}{x^2}\csc^2\left(\frac{1}{x}\right)$?

It's known that $\int\csc^2(x)dx = -\cot(x) + C$, but I don't know how to integrate $\int\frac{1}{x^2}\csc^2(\frac{1}{x})dx$. Can you help? Answer to integral ...
0
votes
2answers
41 views

AP Calculus BC - Related Rates Problem

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
votes
1answer
5 views

How would you plot this equation (in the complex plane)?

So I am currently looking at how to calculate the integral of a complex function $f(z)$ within a contour $\gamma$. That is, an integral of the form $$ \int_{\gamma} f(z) \; dz $$ Where the contour is ...
0
votes
1answer
19 views

Existence of Solution to Integral Equation

How do I show that the integral equation \begin{equation*} x(t) = \ln(1+t) + 1/2\int_0^1e^{-t}\sin^2(ts)x(s)ds \end{equation*} has a solution $C[0,1]$?
0
votes
1answer
21 views

Closed complex integral in an annulus

I have the function $$f(z)=\frac{(e^z-1)(1-\cos(2z))}{z^4\sin(z)},$$ and I want to find $$\oint_{|z|=1}f(z)dz.$$ What I know:  Let $A=\{z\in\mathbb{C}|r<|z|<R\}$ be the annulus with ...
0
votes
1answer
8 views

Sign of a flux surface integral

Use a parametrization to find the flux $$\iint_S F \cdot n \, d\sigma$$ across the surface in a given direction: $$F=xy\overrightarrow i-z\overrightarrow k$$ outward (normal away from the z-axis) ...
1
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0answers
22 views

How to find out the the minimum value of the given integral?

What is the value of $min_{f\varepsilon D} \int_{0}^ {1} (1+x^2)f^2(x)dx$ where $D$={$f:[0,1] \to \mathbb R : f continuous, \int_{0}^{1} f(x)dx =1$} I have no idea how to look for minimum value? ...
3
votes
3answers
380 views

A primitive function of $ e^{x^{2}} $

I made some efforts to set a closed form of primitive function of $ e^{x^{2}} $ i find this function : $ f(x)=\frac{x}{2x^{2}-1}e^{x^{2}} $ where : $f'(x)=(\frac{x}{2x^{2}-1}e^{x^{2}})'$= ...
0
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0answers
13 views

Convergence of indefinite integral

(1) Convergence of $\int_{-\infty}^{\infty}f(x)dx$. (2) Convergence of $\int_{-\infty}^{\infty}f^{2}(x)dx$ What is the relationship betweeen the two? (i,e is there any logical relations...?) (1 ...
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votes
0answers
17 views

Limited integration [on hold]

Let $U_{tt}=4uxx$ with the condition: $U(x,0)= 0 $ $U_t (x,0)= 1$ when $-1<x<1$ and $0$ otherwise Integration from $(x_2t)$ to $x+2t$ ( limits of integration ) for $g(z) dz $, $g(z)= 1$ ...
0
votes
2answers
31 views

Definite integral of a positive continuous function equals zero?

Let's calculate $$\int_0^{\frac\pi 2} \frac {dx}{\sin^6x + \cos^6x}$$ We have $$\int \frac {dx}{\sin^6x + \cos^6x} = \int \frac {dx}{1 - \frac 34 \sin^2{2x}}$$ now we substitute $u = \tan 2x$, and get ...
0
votes
2answers
61 views

Antiderivative of y = $\dfrac {x+22} {x^{2}+2x-8}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
votes
1answer
19 views

Suppose $f:[a,b] \rightarrow [0, \infty)$ is bounded.Let $A(g)=\frac{1}{b-a}\int_{a}^{b}g(x)dx$

Suppose $f:[a,b] \rightarrow [0, \infty)$ is bounded.Let $A(g)=\frac{1}{b-a}\int_{a}^{b}g(x)dx$ for any bounded function $g:[a,b] \rightarrow [0,\infty)$.Show that $A(f)^2 \le A(f^2)$. I was thinking ...
0
votes
0answers
15 views

Using integration to find volume of a parabolic prism.

Suppose I have a solid with its profile being a parabola given by the equation $y = f(x)$. Its depth at any point is given by $g(x)$ and its width is a constant $k$. This essentially results in a ...
87
votes
21answers
8k views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
0
votes
3answers
44 views

Find area of shaded area in curve with range of values for $y$

The parabola in the diagram has equation $y = 32 - 2x^2$ The shaded area lies between the lines $y=14$ and $y=24$ Looking at the graph, I only need to find half the area and multiply by ...
6
votes
3answers
144 views

How do I go about solving this?

I have tried substitution, but it is not working for me. $$ \int_0^\pi \frac{dx}{\sqrt{(n^2+1)}+\sin(x)+n\cos(x)}=\int_0^\pi \frac{n dx}{\sqrt{(n^2+1)}+n\sin(x)+\cos(x)}=2 $$ General form of this ...
1
vote
2answers
18 views

Mistake while evaluating the gaussian integral with imaginary term in exponent

I am trying to evaluate the integral $I=\int_0^\infty e^{-ix^2}\,dx$ as one component of evaluating a contour integral but I am dropping a factor of $1/2$ and after checking my work many times, I ...
1
vote
1answer
114 views

Show that $\int_0^1\frac{1+x^8}{1+x^{10}}dx=\frac{\pi}{\phi^5-8}$

Prove that, $$ \int_0^1\frac{1+x^8}{1+x^{10}}dx=\frac{\pi}{\phi^5-8} $$ What kind of subsititution should be used to solve this integral Another integral that give the same answer but with a ...
-2
votes
0answers
27 views

Confusion related to derivative under integral [on hold]

How to prove that derivative w.r.t $\alpha $ $f(\alpha)=\int_0^{\infty}e^{-x^2}\cos(\alpha x)\ \mathrm dx$ is $-(\alpha/2)$$f (\alpha) $ ? I never did any sum like this.Can someone tell the method to ...
0
votes
2answers
57 views

Which values $a,L$ satisfy $\frac{\int_0^{4π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt}{\int_0^{π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt}=L$?

Which option(s) below have the values of $a$ and $L$ that satisfy the following equation? $$\frac{\int_0^{4π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt}{\int_0^{π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt}=L$$ ...
1
vote
1answer
25 views

How do these partial derivative and derivative terms relate?

From the top line, this proof jumps to the integration and evaluation of the function. I'm not sure how the partial of $t$ and $dt$ play in the integration to give $(s,t)$ before evaluation. Any help ...