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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6
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1answer
185 views

A counterexample

Let $f:\mathbb{R}\to(0,\infty)$ a locally integrable function. I want to compare these two conditions $$\limsup_{r\to + \infty}\frac{r}{\int_{-r}^r f(x)dx}<+\infty. \tag{1}\label{1}$$ and ...
4
votes
3answers
241 views

Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$

Hi I am trying to prove this $$ I:=\int_0^{\pi/4}\log\left(\tan\left(x\right)\right)\, \frac{\cos\left(2x\right)}{1+\alpha^{2}\sin^{2}\left(2x\right)}\,{\rm d}x ...
1
vote
2answers
57 views

How to integrate the following, and what do the results signify?

$$\int\frac{1}{(a^2-x^2)^n}dx$$ Our teacher told us to simply memorize the following answer to this, $a\sin{\theta}$ or $a\cos{\theta}$. But how is it actually derived, and as I missed that class, ...
3
votes
3answers
61 views

Find dimension of the intriguing vector space

We are given vector space of polynomials over $\mathbb R$ of two variables with powers not higher than 2013. Let's consider subspace $V$ which contains such polynomials $f$, so following holds for ...
4
votes
2answers
73 views

2 calculus questions with integration - check me

I have 2 questions I would like assistance with. 1) Find the area of the region bounded by the graphs $y=5x, y=15x, y=\frac{4}{x}, y=\frac{8}{x}$ This was very difficult and tedious. I had trouble ...
11
votes
6answers
241 views
2
votes
0answers
28 views

Greens function of a uniformly charged sphere

The potential $\phi(\boldsymbol{x})$ satisfies $\nabla^2\phi=f$ It may be shown that by defining an appropriate Green's function $g(\boldsymbol{x},\boldsymbol{\xi})$ that ...
0
votes
2answers
44 views

Calculate the energy in a circuit containing a resistor

A voltage peak in a circuit is caused by a current through a resistor. The energy E which is dissipated by the resistor is: Calculate E if Can anyone please give me some suggestions where to ...
0
votes
2answers
52 views

Prove Lipschitz function composed with an integrable function is integrable on [a, b]

Given a Lipschitz function $g$ (i.e. $|g(x) - g(y)| \leq L |x - y|, \forall x, y \in dom(g)$), and an function $f$ integrable on $[a, b]$, how do we prove $g \circ f$ is integrable on $[a, b]$, ...
1
vote
0answers
45 views

The Fourier sine transform of $f(x)/\sin x$

Is the following result $$\lim_{\lambda \to \infty} \frac{2}{\pi} \int_0^\infty \frac{f(x)}{\sin x} \sin(\lambda x) \, dx = f(0) + 2\sum_{k = 1}^\infty f(k\pi),$$ where $\lambda$ is an odd integer, ...
4
votes
6answers
431 views

How should I solve integrals of this type?

The general form of the integral I want to solve is: $$ \int e^{bx}\sin(ax) dx$$ Euler's formula has a nice connection, but the i makes it too complicated. Doing it by parts doesn't seem to get me ...
3
votes
3answers
39 views

Integral properties

**Hi, someone knows how make the procedure to reach, this, I dont undestand how integration limits are change **
0
votes
2answers
51 views

integral of a product of functions being $0$

Suppose we have a continuous function $f$ on $[a,b]$ such that for all integrable functions $g$ such that $\int_{[a,b]}g=0$, $\int_{[a,b]}fg=0 $. Show that $f$ must be constant. Well, it's clear ...
1
vote
3answers
63 views

Integral $\int\frac{e^x \sin x}{\sinh x+\cosh x}dx$

$$\int\frac{e^x \sin x}{\sinh x+\cosh x}dx$$ I know that the answer simplifies to $-\cos x$, but I have no idea how to start this question. Would I use integration by parts? u-substitution?
3
votes
1answer
70 views

Calculating Integral help

I am currently trying to calculate this integral but it is giving me quite a bit of trouble. Would you guys mind giving me suggestions? $$ \int_1^n \int_1^{\sqrt{y}} \frac{1}{x^4 + y^2} dx \, dy$$ ...
1
vote
1answer
19 views

Volume of a Bounded Solid Region Rotated About x-axis

This is a what I think the area bounded between y=x^4 and y=1 rotated about y=4 looks like in 2D. To find its volume, I've also graphed it in 3D. Something I've tried: find the volume of the purple ...
0
votes
1answer
21 views

Calculating the volume of a solid using triple integration

The task says to compute the volume of the solid $A$ created when a cylinder $x^2+y^2 \leq R $ cuts through a ball $x^2+y^2+z^2 \leq 4R^2$, where $A=\{(x,y,z)\in \mathbb R | x^2+y^2 \leq R, ...
3
votes
2answers
68 views

Integral using spherical coordinates

I am trying to compute the volume of the following set : intersection of cylinder $x^2 + y^2 \leq R$ and sphere $x^2 + y^2 + z^2 \leq 4R^2$. I am having trouble setting up the integral properly ...
1
vote
2answers
81 views

Can we prove the formula for surface of revolution?

This is math. We like to prove things. However, proofs are rigorous processes (for a good reason) and are more than just "that idea looks like it could make sense". I've seen proofs for many ...
0
votes
1answer
86 views

Difficult integration problem (unique headline)

So I found this difficult integral at Physicsforum (I'm quite sure you are familiar with this forum) and I've been at it for about an hour now and it's quite frustrating because I can't get the ...
10
votes
7answers
560 views

Beautiful Indefinite Integrals. [closed]

These are some of the integrals with beautiful solutions I came across- $$\int \frac{x^2}{(x\sin x+\cos x)^2} dx$$ $$\int\frac {1}{\sin^3x+\cos^3x} dx$$ $$\int \frac{1}{x^4+1}dx$$ I'd love if you ...
0
votes
1answer
12 views

Find the surface integral of some ellipsoid?

I got Stokes theorem all warmed up for this one! $$\int_{S}\int(Curl(\vec{F}))d\vec{s}$$ (That means delta cross F or curl of F) Where S is the ellipsoid $x^2 + y^2 + 2z^2 = 16$ And $\vec{F} = ...
6
votes
1answer
78 views

Integrate $I=\int_{-1}^3\frac{\sqrt{x+5}}{(1+\sqrt{2x+3})^2}dx$

Compute $$ I=\int_{-1}^{3}\frac{\sqrt{\vphantom{\large A}\,x + 5\,}\,} {\left(1 + \sqrt{\vphantom{\large A}\,2x + 3\,}\,\right)^{2}}\,{\rm d}x $$ Here's what I have tried: Let $t = 1 + ...
5
votes
2answers
131 views

integrate $\tan^2x \sin x$

How do I integrate (find the primitive to) $$ \int \tan^2 x \sin x dx$$ My approach has been to rewrite it to $$\int \frac{\sin^3 x}{\cos^2 x}dx$$ and the do the substitution $t = \tan x/2$. This ...
6
votes
3answers
216 views

Evaluation of $\int_0^1 \frac{\log^2(1+x)}{x} \ dx$

One of the ways to approach it lies in the area of the dilogarithm, but is it possible to evaluate it by other means of the real analysis (without using dilogarithm)? $$\int_0^1 \frac{\log^2(1+x)}{x} ...
1
vote
0answers
75 views

Expectation of $\cos(\|X\|)$ where $X \sim \mathcal{N}(\mu,\Sigma)$

Do: $$ \int_{-\infty}^\infty \int_{-\infty}^\infty \cos\left(\sqrt{x^2+y^2}\right) e^{-\frac{1}{2}\left[\frac{(x-\mu_x)^2}{\sigma_x^2} + ...
0
votes
6answers
87 views

Find a primitive to $\int \frac{\ln x}{x} x^{10} \:\mathrm{d}x$

While battling a bigger problem I encountered this integral that I cannot solve. Which techniques are useful in this (and similar) case(s), and what's the calculations to solve it? $$\int \frac{\ln ...
1
vote
0answers
27 views

Solve a problem of convergence of integral

We have F $\in$ $C(\mathbb{R}^N;\mathbb{R})$, $F\ge0$ and we have that $\int_{\mathbb{R}^N}Fdx<+\infty$. How can i prove the existence of a sequence $r_k\to+\infty$ such that $r_k\int_{\partial ...
2
votes
1answer
90 views

Please help me evaluate this integral

Please help me evaluate this integral: $$\int_\frac{\pi}{4}^\frac{\pi}{2}e^x(\log\sin x+\cos x)\mathrm{d}x$$
1
vote
2answers
51 views

How do I evaluate this definite integral which blows up at lower limit?

I have an integral of the form $$\int^{\infty}_{0}{\frac{2a^2-x^{2} }{a^{2}+x^{2}}e^{\frac{-x^{2}}{b^2}}xdx}.$$ On substitution of $x^2=t$ and simplifying, I get integral of the form ...
14
votes
6answers
490 views

Integral $\int_0^1 \log \frac{1+ax}{1-ax}\frac{dx}{x\sqrt{1-x^2}}=\pi\arcsin a$

Hi I am trying to solve this integral $$ I:=\int_0^1 \log \frac{1+ax}{1-ax}\frac{dx}{x\sqrt{1-x^2}}=\pi\arcsin a,\qquad |a|\leq 1. $$ It gives beautiful result for $a=1$ $$ \int_0^1 \log ...
4
votes
4answers
95 views

Trig integral with sine and cosine

What sort of formulas can I use to reduce this into something I can work with? $$3a^2\int_{0}^{2\pi} \sin^2(\theta)\cos^4(\theta) \, d\theta$$
0
votes
1answer
45 views

volume between the planes $z = ax + by$, $z=0$ and the cylinder of radius 1

Find the volume between the planes $z = ax + by$, $z=0$ and the cylinder of radius 1, whose axis of symmetry is the z axis in the first octant $0≤x, 0≤y, 0≤z$. I´m studying integral in 3 variables, ...
0
votes
1answer
525 views

TI Nspire CX CAS fails to perfrom basic integration

I plugged rather basic integral into my new calculator (TI Nspire CX CAS) just to see what it could do. Surprisingly, it returned undef and I'm wondering if ...
0
votes
0answers
59 views

understanding Green's theorem Intuition

The idea of it is to find the area of a region, yet I keep seeing vector fields popping up all over the place. Take this example from my text book: Find the region enclosed by the two graphs: $y = ...
4
votes
2answers
92 views

Why does Stokes theorem apply to this situation?

I'm thinking Green's theorem or stokes theorem, but I don't know. It has been driving me crazy all day. Help me out here! And if you don't want to help because you know it's homework, give me some ...
1
vote
1answer
203 views

Finding the volume of a cone by integration of parabolic conic sections

I am working on a purely academic way of finding the volume of a right circular cone of height $h$ and radius $r$, (assume $h > r$), using integration of parabolic conic sections (conic sections ...
2
votes
2answers
54 views

Inequality using integration by parts.

I have shown easily that for a function $f$ such that $f(a)=f(b)=0, \int_{a}^{b}f^2=1$, that: $$\int_{a}^{b}xf(x)f'(x)=-1/2$$ But how can I now show that: $$\int_{a}^{b}f'(x)^2 ...
0
votes
5answers
72 views

Computation of an integral using change of variables

So the integral I am trying to compute is: $$\int\limits_B \exp(x+y)dxdy$$ where $B=\{(x,y)|\in \mathbb R| |x|+|y| \leq 1 \}$ Help will be very much appreciated. Thanks!
0
votes
1answer
44 views

Find the area bounded by the hypocycloid?

I have the answer. The hypobloid has parametrization = $x = acos^3(t)$ $y = asin^3(t)$ The explanation is you take a vector field $F(x,y) = (0, x) which has curl 1 than it says the area is equal to: ...
0
votes
1answer
45 views

Can this line integral problem be solved with Stokes theorem?

I have a feeling it could, or with some other theorem. $F(x,y,z) = (2xyz + \sin x)i + (x^2z)j + (x^2y)k$ $$\int_{c} F.ds$$ where $c(t) = (\cos^5(t),\sin^3(t),t^4)$ I tried it in differential form ...
2
votes
0answers
42 views

Is there a way besides integration by parts to solve this integral?

$$\int_{0}^{2\pi} -10\cos^9(t)\sin^4(t)t^4\,dt$$ Maybe a formula for this form or something?
0
votes
0answers
20 views

Line integral: $F(x,y) = (y, x^2 + y^2)$

Calculate $\int_{\gamma} F(x).dx$, for $F(x,y)=(y, x^2 + y^2)$ where $\gamma$ is th arc of circunference $\gamma(x) = (x, \sqrt{4-x^2})$ linking (-2, 0) to (0,2) My attempt $\gamma(t) = (2\cos t, 2 ...
0
votes
2answers
120 views

Integral of $\sin^n(x)$, recurrence relation, some properties

Practicing the manipulation of recurrence relations, I'm stuck on this : Defining $I(n)=\int_{0}^{\pi/2}sin^n(x)dx$, I got the recurrence relation $nI(n)=(n-1)I(n-2)$ for $n\ge2$. Now I'm also ...
1
vote
0answers
145 views

Limit of an integration formula

Let $f$ be a smooth real (or complex) valued function defined on $S^2$. Then a direct calculation shows that $$\int_{S^2}f(x)e^{ixy}\, ...
1
vote
1answer
34 views

Evaluating a line integral through a vector field in 3 dimensions.

Let $\mathbf{F}(x,y,z) = (2xyz + \sin x)\mathbf{i} + (x^2 z)\mathbf{j} + (x^2 y)\mathbf{k}$. Evaluate the integral of $\mathbf{F}$ along $c$, where $c(t) = (cos^5(t), sin^3(t), t^4)$, $t \in [0, ...
5
votes
2answers
108 views

Evaluating the following integral: $\int\frac1{x^3+1}\,\mathrm{d}x$

How to integrate $$\int\frac1{x^3+1}~\mathrm{d}x$$ Is it possible to use Taylor expansion?
1
vote
1answer
97 views

Evaluating an integral.

I am trying to integrate $$\iint_{D}^\ y^3(x^2+y^2)^{-\frac{3}{2}} dx\, dy$$ on $D$, the region determined by the conditions $1/2\leq y\leq 1$ and $x^2+y^2\leq 1$ My book suggested we use $y^2=u$, ...
3
votes
2answers
103 views

$I=\int_1^\infty \frac{dx}{x\sqrt{x^2-1}}$Integrating this

$$ I=\int_1^\infty \frac{dx}{x\sqrt{x^2-1}}=? $$ Thank you. I dont know how to do it because the bounds on the integral, I feel like that is confusing me a lot more than it means to be. I tried ...
4
votes
1answer
94 views

Evaluate $\int_0^\infty e^{-b\left(\frac{r^2}{a^2}+1\right)^\frac{-\gamma}{2}}\left (\frac{r^2}{a^2}+1\right)^\frac{-\gamma}{2} r^2 dr $

I'm working on research in astrophysics related to determining the ages of stellar nurseries. I've got the numerical solution, but need an analytic solution to the integral below in order to better ...