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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21 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
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1answer
31 views

calculus use of integral calculus [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 accrued from ...
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3answers
85 views

Evaluate $\int_0^\infty\frac{\mathrm{d}l}{(r^2+l^2)^{3/2}}$

How to evaluate the following integral $$\int_0^\infty\frac{\mathrm{d}l}{(r^2+l^2)^{3/2}}$$ The solution is supposed to look like this, unfortunately I can't derive it. $$ ...
2
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3answers
163 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
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3answers
45 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 ...
-3
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0answers
28 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 ...
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1answer
29 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 ...
8
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1answer
104 views

A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
1
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1answer
58 views

Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right) $ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
-3
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1answer
28 views

Solve using the integration by parts method [on hold]

Greeting Tutor, I having a trouble to solve attached equation using integration by parts method, appreciate tutor help to provide guide or step. Million thanks.
2
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1answer
41 views

Computing double integral

Find $$\iint\limits_D \sqrt{(x-10)^2+y^2}\hspace{1mm}dA$$ where $\{(x, y)\in D \mid x^2+y^2\leq 10^2\}$. I am not sure how to start, every way I have tried so far, ends up into something ugly. All ...
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5answers
96 views

Definite integral $\int_{-\pi/2}^{\pi/2}\cos^{2}\left(\theta\right)\,{\rm d}{\theta} $

Please help me to evaluate definite integral $$\int_{^{-\pi}/_2}^{^\pi/_2}\cos^{2}\left(\theta\right)\,{\rm d}{\theta}$$ Also there was a hint: Use the double angle formula ...
4
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1answer
73 views

Reverse Cauchy Schwarz for integrals

Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that ...
1
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1answer
30 views

Finding a function from a vector field

The vector field $F(x, y) = \left(\displaystyle\frac{x}{r^3}, \frac{y}{r^3}\right)$ appears in electrostatics, where $r = \sqrt{x^2 + y^2}$ is the distance to the charge. Find a function $f(x, y)$ ...
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0answers
27 views

Definite Integration.Trigonometric function [on hold]

How to integrate $$3\sqrt { \cos ^{ 2 }{ \left( t \right) \sin ^{ 2 }{ \left( t \right) +\sin ^{ 4 }{ \left( t \right) \cos ^{ 2 }{ \left( t \right) } } } } } $$ for $t\epsilon \left[ 0,2\pi ...
0
votes
0answers
31 views

Laplace transform of product of two function

Consider the following integral of product of two function $$ \int_0 ^t f(s)g(s-t)ds $$ i want to know the laplace transform of above term w.r.t t. if $g(s-t)$ is replaced by $g(t-s)$, there is a ...
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3answers
95 views

Value of the integral $\int_{\mathbb{R}} \frac{x\sin {(\pi x)}}{(1+x^2)^2}$

How do we evaluate the integral $$I=\displaystyle\int_{\mathbb{R}} \dfrac{x\sin {(\pi x)}}{(1+x^2)^2}$$ I have wasted so much time on this integral, tried many substitutions $(x^2=t, \ \pi x^2=t)$. ...
1
vote
3answers
82 views

Evaluate $\int_{1}^{e}\frac{u}{u^3+2u^2-1}du.$

I'm trying to solve $$\int_{1}^{e}\frac{u}{u^3+2u^2-1}du.$$ My first approach was to factorise and then do a partial integration. However the factorisation ...
6
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0answers
95 views

An incorrect answer for an integral

As the authors pointed out in this paper (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed). $$ ...
3
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2answers
232 views

What is the value of this double integral?

Let $C$ be the subset of the plane given by $$ C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | \ 0 \leq x^2 + y^2 \leq 1 \}.$$ Then what is the value of the double integral $$ \int_{C} \int (x^2 + y^2) ...
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2answers
56 views

How to evaluate this double integral?

Let $C$ be the subset of the plane given by $$C \colon= \{ \ (x,y) \in \mathbb{R}^2 \ | -1 \leq x = y \leq 1 \}. $$ Then how to evaluate the double integral $$ \int_C \int (x^2+ y^2) dx dy? $$ My ...
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1answer
35 views

Partial fraction for integrating

I have been trying to solve the integral $\displaystyle\int \frac{dx}{(x-1)^2 (x^2+1)^3}$. So while trying to get the partial fraction which way is better? ...
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0answers
23 views

Regarding methods of finding a derivative.

I read in the American Mathematical Monthly Descartes found away to calculate the slope of a tangent to a curve at a point specified. Called the Double tangent point method ( I think). This method ...
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2answers
40 views

What is the area bounded by these curves?

Let $f(x) \colon = x^2$, $g(x) \colon= x+1$. Then what is the area bounded by the graphs of $f$ and $g$ between the vertical lines $x= -1$ and $x= (1+\sqrt{5})/2$? My effort: Since $$ f(x) - g(x) ...
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2answers
45 views

Integration of piecewise defined function: $ f(x)=0$ for $x<1$ and $f(x)=1$ for $x\geq1$

I think I am confusing myself too much on this. Let $ f(x)=0$ for $x<1$, and $f(x)=1$ for $x\geq1$. What is $\int_0^1f(x)\,dx$? I am worried because $f$ is discontinuous at $1$. Does that make ...
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1answer
39 views

Eigenvalues and Eigenfunctions of Integral Equation

Compute the eigenvalues and eigenfunctions of $$ \varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x) $$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...
0
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1answer
22 views

Rope question - integration

A 50-lb bucket is at the bottom of a 100-ft well. A 200 lb rope (also 100 ft long) is tied securely to the bucket. We will use rope to lift this bucket out of the wall, at a rate of 1 foot every ...
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1answer
38 views

Finding volume of a solid

Okay so this question comes in two parts and the second part of the question doesn't make any sense to me. Let $R$ be the region in the first quadrant that lives between the curves $f(x) = x^2$ and ...
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0answers
35 views

Passing of the limit for Lebesgue Integral (Proof Verification)

Let $f_n\in L^1(0,1)$ and $C>0$ be such that $f_n \geq 0, f_n \rightarrow 0$ a.e., and $$\int_0^1 \max\{f_1, ..., f_n\} dx \leq C \quad \text{ for every } n.$$ Prove that $f_n \rightarrow 0$ in ...
2
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2answers
89 views

Wrong interpretation of the indefinite integral

This might sound very useless but I'd like to see what you think. Bear in mind that I'm just a novice student. if $f$ is the original function, then it could be found this way $C+\int f'(x)\, ...
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0answers
24 views

Show that different eigenfunctions of integral kernel are orthogonal

Consider the integral operator $K \varphi := \int_0^1 k(x,s) \varphi(s) ds$ with a continuous and symmetric kernel $k : [0,1]^2 \to \mathbb R$ which has at least two different eigenvalues $\lambda_1$ ...
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4answers
471 views

How useful/useless is the indefinite integral

After having met yet another person confused by indefinite integrals today, I've finally decided to ask the community. Do you think it makes sense to teach indefinite integrals? My opinion is that ...
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2answers
151 views

An exercise from my brother: $\int_{-1}^1\frac{\ln (2x-1)}{\sqrt[\large 6]{x(1-x)(1-2x)^4}}\,dx$

My brother asked me to calculate the following integral before we had dinner and I have been working to calculate it since then ($\pm\, 4$ hours). He said, it has a beautiful closed form but I doubt ...
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0answers
35 views

Integral of Hypergeometric Function with polynomial and exponential

I was working on some mathematical derivations and faced this integral: how can I integrate it?
8
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1answer
132 views

Integral: $\int_0^{\pi} \frac{x}{x^2+\ln^2(2\sin x)}\,dx$

I am trying to solve the following by elementary methods: $$\int_0^{\pi} \frac{x}{x^2+\ln^2(2\sin x)}\,dx$$ I wrote the integral as: $$\Re\int_0^{\pi} \frac{dx}{x-i\ln(2\sin x)}$$ But I don't find ...
0
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1answer
15 views

Proving some properties about the expected first order statistic (maximum) with respect to sample size.

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as ...
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0answers
55 views

Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
13
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1answer
201 views

Prove ${\large\int}_0^\infty\left({_2F_1}\left(\frac16,\frac12;\frac13;-x\right)\right)^{12}dx\stackrel{\color{#808080}?}=\frac{80663}{153090}$

I discovered the following conjectured identity numerically (it holds with at least $1000$ digits of precision). How can I prove it? ...
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2answers
43 views

Volume of trig function around y-axis

I have this question and it's the first kind of question I'm doing involving finding volume so I just would like some help solving this question: Find the volume created by revolving the curve $ \ ...
4
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3answers
123 views

Evaluate $\int_0^1\frac{x^a-x^{-a}}{x-1}dx$

I have heard that: $$\int_0^1\frac{x^a-x^{-a}}{x-1}dx=\frac1 a-\pi\cot(\pi a)$$ when $-1<a<1$. How would I prove this? That doesn't have an elementary indefinite integral, but the definite ...
11
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0answers
180 views

Evaluating $\int_0^\pi\arctan\left(\frac{\log\sin x}{x}\right)\mathrm{d}x$

I found the following integral as a by product of another one. It has a nice closed form. $$ \displaystyle \int_{0}^{\pi} \arctan \left( \frac{\ln \sin x}{x} \right) \mathrm{d}x $$ ...
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1answer
42 views

Area of solid revolution using integration.

When we calculate the volume of a solid generated by rotating a curve around $x$-axis, We divide it into disks. So ,we get $dv = \pi r^2 dx$. where $r=y$ and then we integrate. That OK, but when ...
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1answer
17 views

Average value for multiple integrals

If there is a function $f(x,y)$ and we want to find the average value over a region $R$ defined by $0<x<1$ and $0<y<x$, how is that computed? I know that it would be something like this: ...
3
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1answer
37 views

Generalising integration by parts for the product of more than two functions

Just as the product rule can be generalised to the product of more than two functions, i.e. $$\frac{d}{dx} \left [ \prod_{i=1}^k f_i(x) \right ] = \sum_{i=1}^k \left(\frac{d}{dx} f_i(x) \prod_{j\ne ...
3
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2answers
74 views

Intuition about Taking an Integral

My hope is to personally develop some further intuition for taking an integral (measuring the area under a curve). Consider a normal distribution and I need the area under the curve from $a$ to $b$. I ...
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0answers
57 views

Is the following integration of possible?

How to solve the following problem? $\int x^4/(1-x^4)^{3⁄2}dx$ I have tried the substitution, x=sinz, but failed.
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0answers
16 views

Volume of Solid of Revolution

This problem is giving me some trouble: The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. x = (y − 5)^2, x = 4; ...
1
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0answers
41 views

Integrating $\int_0^1 dx\,\ln(x-a)/(x-b)$ paying attention to cuts.

I am trying to compute the following integral, for complex $a$ and $b$ $$\int ^1 _0 dx \frac{\ln(x-a)}{x-b}$$ by turning it into something in terms of dilogarithms. But for certain values of $a$ ...
0
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1answer
38 views

Use triple integrals to integrate over a tetrahedron

Integrate $f(x, y, z) = x^2 + y^2 - z$ over the tetrahedron with vertices $(0, 0, 0), (1, 1, 0), (0, 1, 0), (0, 0, 3)$. I need to use triple integrals to solve this, so I made a diagram and set the ...
0
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1answer
60 views

Integral of the exponential function

I am searching the indefinite integral of this function: $\dfrac{\exp(x)}{(1+x)^{5/3}}$. Thank you alot.