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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8
votes
1answer
152 views

Interesting integral: $I=\int_0^1 \int_0^1 \log\left( \cos(\pi x)^2 + \cos(\pi y)^2 \right)dxdy$

I've stumbles across the following integral when doing some combinatorial work: $$ I=\int_0^1 \int_0^1 \log\left( \cos(\pi x)^2 + \cos(\pi y)^2 \right)dxdy$$ After plugging this into Mathematica, it ...
0
votes
1answer
20 views

An integral related to the harmonic number

Definitions $\Gamma(s)$ is a Gamma function Defined as $\Gamma(s+1)=s!$ $H_{n,s}=\sum_{k=1}^{n}\frac{1}{n^s}$ We proposed: proof that, $$-\frac{1}{\Gamma(s)}\int_0^\infty x^{s-1}\cdot ...
0
votes
3answers
102 views

the relationship between $f(x)$ and $dx$ in $\int_a^b f(x)\,\mathrm{d}x$

In this example, If we use the relation $F'(x) = f(x)$, this consequence of the fundamental theorem may be written in the form $$ f(b) - f(a) = \int_a^b F'(x) dx = ...
3
votes
2answers
147 views

How to evaluate $\int_0^\infty \frac{e^{-x}+x-1}{x(e^{2x}-e^{-2x})}dx$? [closed]

We are unable to verify this this equality $$ 4\int\limits_0^\infty \frac{e^{-x}+x-1}{x\left(e^{2x}-e^{-2x}\right)}\;\mathrm{d}x=\gamma+\ln\frac{16\pi^2}{\Gamma^4\left(\frac{1}{4}\right)}\;. $$ ...
0
votes
0answers
48 views

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

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

Volume of paraboloid that is cut with plane

How to calculate the volume of the paraboloid : x^2 + y^2 = z that is cut with x + y + z = 5 plane. Please give several methods if you can. Thank you very much for answers.
1
vote
1answer
32 views

Norm square of an integral

Is it allowable to do like this: $$ \Psi(x,t) = \int_{-\infty}^{+\infty} e^{k^2/a}*e^{ikx}*e^{-ik^2t} $$ $$ |\Psi(x,t)|^2 = |\int_{-\infty}^{+\infty} e^{k^2/a}*e^{-ikx}*e^{-ik^2t}dk|^2 $$ $$ ...
0
votes
0answers
19 views

Quadrature over a (smooth, compact, convex, etc.) Riemannian manifold

Problem setting Consider three points on the surface of the earth (which I want to assume to be a perfect ellipsoid here) that are pairwise sufficiently close for unique geodesics to be found between ...
-6
votes
2answers
45 views

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

Problem 3: Find $$\int xe^xdx$$ by using appropriate integration techniques. [5 marks]
3
votes
1answer
634 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 + ...
0
votes
1answer
43 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
votes
0answers
16 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
votes
1answer
47 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
2answers
32 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
votes
1answer
374 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 ...
4
votes
2answers
58 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. ...
2
votes
1answer
497 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
952 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 * ...
1
vote
4answers
102 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
votes
1answer
29 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
vote
1answer
32 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
52 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
60 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
26 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
vote
1answer
21 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 ...
2
votes
1answer
99 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? $$ ...
-1
votes
2answers
79 views

Easiest way to solve this integral [closed]

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
28 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 ...
3
votes
3answers
62 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
95 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
49 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
7 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
26 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
23 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
11 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
vote
0answers
26 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
381 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
votes
0answers
15 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 ...
-2
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
32 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
64 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
20 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
19 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 ...
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
157 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
27 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
121 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
28 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
58 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
28 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 ...