0
votes
1answer
10 views

Question about Riemann Integration and the indicator function

Let $S \subseteq \mathbb{R}^n$. Suppose $\chi_S$ is integrable and $\int_Q \chi_S = 1 $ for some rectangle $Q$ such that $S \subseteq Q $. Let $\epsilon > 0 $ be given, I want to ask how can I ...
0
votes
1answer
36 views

How to prove that $\max\{f,g\}$ is Riemann integrable? [duplicate]

If f(x) and g(x) are Riemann integrable in [a,b], why $h(x)=\max\{f(x),g(x)\}$ is still Riemann integrable in [a,b]? Or maybe it is wrong?
2
votes
1answer
39 views

Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
0
votes
1answer
27 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
-1
votes
1answer
22 views

Laplace's Method (Integration)

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} Use Laplace's Method to show that \begin{equation} I(x) \sim ...
-1
votes
1answer
33 views

Inverse integration of a special definite integral

Hi I am facing a problem with this problem. please help.
2
votes
2answers
100 views

Why do we bother with $u$-substitution?

This question has bothered me ever since I learned $u$-substitution (A note here: I have no formal education at this level, so I may definitely have missed something). The method is presented as an ...
0
votes
2answers
20 views

How to integrate by reduction method

How to evaluate the integrals of (a) $(\ln(x))^n$ (b) $x^ne^{ax}$ where $a$ is a constant By reduction formula
2
votes
1answer
26 views

Re-interpreting double integral as a Type II Region $\mathrm{d}y\,\mathrm{d}x$ vs $\mathrm{d}x\,\mathrm{d}y$

I have the following Double Integral:$\iint_Dx\cos y\space\mathrm{d}A$ where $a$ is bounded by $x=1,y=0,y=x^2$. Interpreting this region as a Type one region, it is easy to conclude $R=\{(x,y)\mid ...
0
votes
3answers
37 views

How to solve integration of $\int x(x^2+k^2)^{-1/2} \, dx$?

As said in title, how do you solve integral $\int x(x^2+k^2)^{-1/2}\,dx$ where $k$ is some constant?
6
votes
2answers
98 views

Compute $I=\int_0^{+\infty}\frac{\arctan(t)}{e^{\pi t}-1}dt$

I would like to compute $\displaystyle I=\int_0^{+\infty}\frac{\arctan(t)}{e^{\pi t}-1}dt$ Let $D=(0,+\infty)$, I have $\frac{1}{e^{-\pi t}-1}=\frac{e^{-\pi t}}{1-e^{-\pi t}}$ So ...
1
vote
1answer
34 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
1
vote
1answer
43 views

Suppose $f(x)\in L_1$ - Prove that $\lim_{n\rightarrow\infty}\int_0^\infty f(x)\cos(nx)dx = 0$

Assuming knowledge of the cyclic behavior of $cos(x)$, integration by parts, and $\int_0^{\infty} f<\infty$ is enough here? Consider \begin{align} & \int_0^\infty f(x)\cos(nx)dx = ...
2
votes
2answers
39 views

Convergence of $\int_{0}^{+\infty}\ln(1+\frac{1}{t^2})$

Study the convergence of $\int_{0}^{+\infty}\ln(1+\frac{1}{t^2})dt$ For $+\infty$ case it's easy we have $\ln(1+\frac{1}{t^2})\sim \frac{1}{t^2}$ For $0$ case I feel it's ...
0
votes
1answer
42 views

Are there integrals you can't solve without inverse hyperbolic substitution?

Are there any integrals that can't be solved with only trig substitution? An integral that requires you to use a hyperbolic or inverse hyperbolic substitution?
1
vote
0answers
24 views

Find the power series for a definite integral

I am a bit unsure when integration is used together with summation. Here is my question: Find power series for $\int_0^{1} \frac{\sin x}{x}dx$ in the form $\sum_{k=1}^{\infty} a_kx^k$ Here is what I ...
2
votes
0answers
23 views

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
1
vote
2answers
81 views

Evaluate $\iiint xyz$

Evaluate $$\iiint_E xyz\, dV$$ where $E$ is the solid: $0\leq z\leq 9,\,0\leq y\leq z,\, 0\leq x \leq y.$ I am having a hard time drawing a picture of this solid $E$ to find out what the ...
2
votes
0answers
27 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
0
votes
1answer
15 views

Parametrize plane and get surface area

Find a parametrization of the surface: $y + 2z = 2$ inside the cylinder $x^2 + y^2 = 1$. Then, compute its surface area. I'm having trouble finding the parametrization of the surface. I don't think ...
0
votes
0answers
16 views

Construction antiderivative analytical

Find the exact area between the curves $y=x^2$ and $y=2-x^2$ with antiderivative analytical method. (fundamental theorem of calculus )
3
votes
1answer
28 views

Tricky looking integration (after separation of variables)?

I've come across something in my notes that jumps from: $${d\rho \over dz} = \sqrt{\left({\rho \over C}\right)^2 - 1}$$ to: $$\rho(z) = C \cosh\frac{z-z_0}{C}$$ I know that separation of variables ...
1
vote
0answers
30 views

Integral of $\sin^{-1}(1/2 - \sin x) dx$

Obviously the solution exists for the integral $\int\sin^{-1}(\sin x) dx$, but does the solution exist for $\int\sin^{-1}(1/2 - \sin x) dx$? Or, for that matter, $\int\sin^{-1}(\alpha - \sin x) dx$ ...
0
votes
0answers
16 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
1
vote
1answer
38 views

Compute $\int_{0}^x \vert \sin(t)\vert dt$ for $x\in \mathbb{R^+}$

Let $x\in \mathbb{R^+}$, compute $$\int_{0}^x \vert \sin(t)\vert dt$$ I tried like this : $$ \int_{0}^x \vert \sin(t)\vert dt=\int_0^{\lfloor \frac{x}{\pi}\rfloor \pi}\vert \sin(t)\vert ...
3
votes
0answers
38 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
5
votes
1answer
57 views

Problem with a sequence with multiple integrals [duplicate]

How to compute the following limit, $\displaystyle \lim\limits_{n \to \infty} \int_0^1 \int_0^1 \ldots \int_0^1 \sin \bigg(\frac{x_1+x_2+\ldots+x_n}{n}\bigg)\,dx_1 \,dx_2 \ldots \,dx_n$ ? I will ...
0
votes
1answer
30 views

Calculate a triple integral - variable changed into spherical coordinates

The problem is to calculate $$\iiint_D x^2\,dx\,dy\,dz$$ where $D$ is determined by $x^2+2y^2+z^2\le2$. solution my attempt: why can I not do it like that? I change variables, calculate the ...
0
votes
1answer
31 views

Have I done something wrong in this integral?

I have showed most of my steps here so I hope that this is easy to follow. I have the integral $$A = C\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}Y^*(\theta, \phi)f(\theta,\phi)sin(\theta) d\theta ...
0
votes
3answers
42 views

How to find $\int\sqrt{(26x-x^2)}dx $

How do I find $\int \sqrt{(26x-x^2)} dx $ Is this an integration by parts question? Thanks, --Nick
6
votes
1answer
120 views

How to prove $\int^1_0\int^1_0\frac{\log(x-x^2)-\log(y-y^2)}{(x-x^2)-(y-y^2)}dxdy=7\sum_{i=1}^\infty i^{-3}$?

How do you prove that $$\int^1_0\int^1_0\frac{\log(x-x^2)-\log(y-y^2)}{(x-x^2)-(y-y^2)}dxdy=7\sum_{i=1}^\infty i^{-3}\;\;\;\left(=7 \zeta(3)\right)~?$$ p.s. Mathematica gives a pretty good ...
2
votes
2answers
46 views

Help changing the order of integration

So I need to change the order of integration. I am giving the following limits, $1 \leq x \leq 9$ and $\sqrt{x} \leq y \leq 4$. I am having no luck solving this one. Any help would be greatly ...
3
votes
2answers
47 views

Double integral help

I'm having difficulty with a question. It says By putting $x=r\cos(\theta), y=r\sin(\theta)$, prove that $$\int_0^{\infty}\int_0^{\infty}e^{-(x^2 + 2xy\cos(\alpha)+y^2)}dx\ ...
-1
votes
0answers
10 views

Integral Evaluation with MATLAB-Mupad (triple and lesser degree integrals)

https://www.wolframalpha.com/input/?i=integral+of+2c%28x%5E2%2By%5E2%29%28√%28a%5E2+-+x%5E2+-+y%5E2%29%29+with+respect+to+y+from+-√%28a%5E2+-+x%5E2%29++to+√%28a%5E2+-+x%5E2%29 Here is a link to the ...
4
votes
0answers
80 views

The long Integral with a nice result

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2Li_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) -\frac{\zeta(2)-2\log^2 ...
1
vote
2answers
82 views

Equality of integrals: $ \int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \int_{0}^{1} \frac {1}{1+x^2} \, \mathrm{d}x $

In Street-Fighting Mathematics (page 16), Prof. Sanjoy Mahajan states that $$ \displaystyle\int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \displaystyle\int_{0}^{1} \frac {1}{1+x^2} \, ...
1
vote
1answer
30 views

Does it Make Sense to Use the Variable of Integration as a Bound?

I can't for the life of me seem to decide if using the variable of integration as a bound makes sense. For instance, integrating $y=x$ from $0$ to $x$. I don't think it does… But I'm not sure.
0
votes
1answer
16 views

Numerical integration: Quadrature method which one to use?

Since “it depends” is the proper answer to a question about what quadrature method to use in evaluating an integral, what are the things that one should consider when making a choice.
1
vote
1answer
36 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
0
votes
1answer
22 views

natural log integral question dx/(13-x)

Just wondering why the answer to the integral: $$\int \frac{\mathrm{d}x}{13-x}$$ is $-\ln|x-13|$ as opposed to $-\ln|13-x|$. Why do the $13$ and $x$ get switched?
0
votes
1answer
44 views

Expansion of Integration

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
0
votes
1answer
19 views

Solving a double integral using substitution

The problem: Evaluate $$\iint_{D}(x+y)^2(x-y)^5\:\mathrm{d}x\:\mathrm{d}y,$$ where $D$ is a rectangle with vertices in $(0, 1), (1, 0), (1, 2), (2, 1)$. So I drew the square and thought up this ...
1
vote
1answer
10 views

Finding all continuous solutions to an integral

I need help with both parts of this problem. Part (i) seems obvious, because the integrand $f(t)$ would become $F(t)$, which is obviously differentiable because its derivative is $f(t)$ by ...
2
votes
2answers
54 views

Integral $I=\int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx$

Hi I am trying to show $$ \int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx=\frac{\pi}{32\alpha^{3/2}},\quad \Re(\sqrt \alpha)> 0. $$ I am looking for a solution to this NOT using contour integration, but ...
3
votes
1answer
61 views

Integral definition of e

I know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...
1
vote
0answers
26 views

Infinitesimal multiplication

Taking the exponential of the sum of logarithms gets you the product of the terms. Now, if we have a function with positive values, we can form $$\exp\int_a^b \ln f(t) dt.$$ In some sense, this is ...
4
votes
1answer
78 views

Solving integral $ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x $

there is integral $$ \int \frac{x+\sqrt{1+x+x^2}}{1+x+\sqrt{1+x+x^2}}\:\mathrm{d}x$$ i am trying to separate this : $$=\int \mathrm{d}x -\int \frac{\mathrm{d}x}{1+x+\sqrt{1+x+x^2}} $$ but have no idea ...
1
vote
0answers
17 views

difference of the values of a function is an integral

This is a very simple quesiton but something I don't understand. From Taylor expansion: $$f(y)-f(x)=f'(x)(y-x)+O((y-x)^2)$$ so, if I just picture that, on the left is the difference between two values ...
3
votes
2answers
47 views

Evaluate $\int\frac{\mathrm{d}x}{2x-4}$

My question is to evaluate: $$\int\frac{\mathrm{d}x}{2x-4}$$ Why is the solution equal to $\frac{1}{2}\ln|x-2|$ as opposed to $\frac{1}{2}\ln|2x-4|$? I understand that if I factor $\frac{1}{2}$ ...
3
votes
2answers
75 views

Evaluate $\int x \sqrt{1 - x^4} \,\mathrm{d}x$

I have the following question $$\int x \sqrt{1 - x^4} \,\mathrm{d}x$$ I know we have to use trig. substitution for this and therefore, I did the following by letting $x = \sin \theta$ and $dx = \cos ...