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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30 views

For u-substitution, what is the mathematical systematic way of picking a u?

What is the systematic way of figuring out what u should be set equal to? In class we were introduced to u-substitution, which is the chain rule in reverse. You pick a u that is the inner function ...
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0answers
49 views

evaluation of fourier transform of electric potential

I would like to ask how to evaluate equation 7? I have spent hours and still have no idea how to get a(k).
3
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2answers
64 views

Transforming the integral $\int_0^\infty e^{-x^2}\sin(x) dx$ into a specific sum

Using the series representation of $\sin x$, I want to prove that: $$\int_0^\infty e^{-x^2} \sin(x) dx = \frac{1}{2} \sum_{k=0}^\infty (-1)^k \frac{k!}{(2k+1)!}$$ My attempt: I've started by ...
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1answer
13 views

Integral with scalar product and norm

I need to show that for $f\colon \mathbb{R} \rightarrow \mathbb{C}$ continuous that the following holds for a fixed $x \in \mathbb{R}^{n+1}$: $$\int_{S^n} f(\langle x, y \rangle) d\mu(y) = \int_{S^n} ...
1
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1answer
43 views

Any continuous function on a closed interval in $\mathbb{R}$ is integrable

Prove that any continuous function on a closed interval in $\mathbb{R}$ is integrable. Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function. We want to show that for any $\epsilon>0$ ...
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0answers
47 views

Computing the limit of this integral,

This is Part 6 (last part) of a problem statement of an old comprehensive exam question that I am working on. It asks to evaluate $$\lim_{r_0 \to 0} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\...
3
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1answer
239 views

Prove that $\int_0^1 f(x)dx=0$ if $f(\frac{1}{n})=1$ for $n=1,2,3,\ldots$ and $f(x)=0$ for all other $x$

Prove that $\int_0^1 f(x)dx=0$ if $f(\frac{1}{n})=1$ for $n=1,2,3,\ldots$ and $f(x)=0$ for all other $x$. Lemma: If $f:[a,b]\rightarrow \mathbb{R}$ is a function such that $f(x)=\mathbb{1}_{\{c\}...
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1answer
59 views

Boundary on $R^3$ about Stoke's theorem.

Let S be the union of two surfaces, $S_1$ and $S_2$, where $S_1$ is the set of $(x,y,z)$ with $x^2+y^2 =1$ ,$0 \le z \le 1$ and $S_2$ is the set of $(x,y,z)$ with $x^2+y^2+(z-1)^2 =1$, $z \ge 1$. Set $...
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1answer
45 views

Is there a means of finding an infinite sum by means of altering it into an integral?

If you are given a sum, say $$\sum_a^b f(x)$$ with $a,b\lt \infty$ Is there a means of solving for this sum by means of integration? (I am familiar with sophomores dream.) Thank you for any help
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1answer
37 views

Fundamentals of applications of limits in integration

I have a very fundamental question and a basic question which I am getting very confused while using the calculator and also while solving the question in the paper. Let me give you an example $$\...
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2answers
69 views

How to show that the following function integrates to a finite value?

I am interested in showing that the following integral evaluates to some finite value. How can I do that ? Note: Outer integral is from $0$ to $1 $ (I forgot to type that in the outer integral for ...
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1answer
23 views

Find the area bounded for $f(x)=x^2-4x$ and $x+y=0$

Find the area bounded the following graph $f(x)=x^2-4x$ and $x+y=0$ I'm not sure how to tackle this problem. The y is kind of throwing me off here. For other problems that I have been doing, there ...
2
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2answers
62 views

How to solve infinite repeating integration.

I am having trouble solving the following integration $$\frac{1}{y!}\int^\infty_0\!x^ye^{-2x}\,\mathrm{dx}$$ We see that in order to solve this, we need integration by parts. $u = x^y$ $du = yx^{y-...
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1answer
45 views

Find the average value of $f(x)=x^2+5$ in $[0,3]$

Find the average value of $f\left(x\right)=x^2+5$ for $x \in [0,3]$. So far using $$\frac{1}{b-a} \int_a^b f(x) dx$$ I got $\frac{1}{3}\left(\frac{x^3}{3}+5x\right)$ so do I just plug in the $0,3$ ...
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2answers
87 views

The function $f$ is Riemann integrable on $[-1,1]$. For any interval in $[-1,1]$, there are always both positive and negative values of $f(x)$.

The function $f$ is Riemann integrable on $[-1,1]$. For any interval in $[-1,1]$, there are always both positive and negative values of $f(x)$. How can I prove that $\int^1_{-1}f=0$? I think that ...
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1answer
46 views

Evaluats this integral [closed]

Evaluate : $$\int _0^1 \frac{5x}{\left(5x^2+6\right)^2} \mathrm{d} x$$ not sure how to solve this problem
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2answers
28 views

indefinite integral by substitution

I know we are supposed to post as far as we get on a problem in order not to waste people's time, and I understand the reason for that. But I really couldn't get very far on this problem. I am not ...
3
votes
2answers
76 views

evaluating an indefinite integral with tangent function

I'm studying for finals, and I'm having a problem with this integral: $$\int\tan(7x+1)dx$$ I simplified it to $$\int \frac {\sin(7x+1)}{\cos(7x+1)} dx .$$ I then substituted $\cos(7x+1)$ by $u$, ...
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2answers
596 views

Find dy/dx for an integral?

Can someone walk me through how to find dy/dx (one of the problems I'm reviewing in my Calculus book): $$\int_{1/x}^{2} t\sqrt{t-4} dt $$ I know I need my (x) ...
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1answer
51 views

Iterated integral of $x/(1+y^3)$

$$\int_0^{10}\int_x^{10}\frac{x}{1+y^3}dydx$$ I assume there's a neat substitution but I can't see one. Using Fubini's theorem doesn't seem to be much help either.
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1answer
29 views

Convolution as a $L^1$ limit of translates.

I would like what convolution is, as a $L^1$ limit. Namely let $f,g\in L^1(\mathbb{R})$ (with some further conditions). Then what conditions on $f$ and $g$ ensure that $f\ast g$ is the $L^1$ limit for ...
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0answers
30 views

$\int_B( x^2+y^2) dxdydz$ where $B=\{(x,y,z)| x^2+y^2\le 1; 0\le z\le 1\}$

I need to take the integral: $$\int_B (x^2+y^2) dxdydz$$ where $$B=\{(x,y,z)| x^2+y^2\le 1; 0\le z\le 1\}$$ so I did like this: first I integrated $x^2+y^2$ over $z$ from $0$ to $1$: $$\int_0^...
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2answers
39 views

Find the mass of a 2-dimensional triangle plate

The triangle has the following vertices: $$(0,0),(c,0),(0,c)$$ I drew this figure and found the slope and the line: $$m=\frac{0-c}{c-0}\frac{-c}{c}=-1$$ And was then able to form the following ...
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1answer
35 views

Proof for definite integral inequality

Prove this inequality $$ \frac\pi4-\frac12\le\int_0^1\frac{\arcsin x}{1+x^8}\,dx<\frac\pi2-1$$ I have worked with the range of inverse sin which didn't give me anything
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1answer
41 views

Indefinite Integral.

We define Indefinite integral of a function $f$ as $\int f(x)dx$ which is the collection of anti derivatives of the function $f.$ But in many books it is written that for an integrable function $f$...
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2answers
44 views

Is the quotient of two continuous functions absolutely integrable?

(True or False) Suppose that $h$ is absolutely integrable on $(a,b)$. If $f$ is continuous on $(a,b)$, if $g$ is continuous and never 0 on $[a,b]$, and if $|f(x)|\leq{h(x)}$ for all $x\in[a,b]$, then $...
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1answer
83 views

Find the volume of a body bounded by $z = x^2 + 3y^2, z= 8-x^2-5y^2$.

Find the volume of the body bounded by the elliptic paraboloids $z = x^2 + 3y^2, z= 8-x^2-5y^2$. What i did? First of all i intersected two functions and got the desire area is a ellipse $x^2+4y^...
3
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0answers
138 views

Is $h(\mathbf{a},b)=\int_\Omega f(\mathbf{x})e^{-g(\mathbf{x})}\mathrm{d}\mathbf{x}$ differentiable?

Let $f\colon\Bbb{R}^n\to\Bbb{R}$ be an affine function and $g\colon\Bbb{R}^n\to\Bbb{R}$ be a non-negative function. We define $h\colon\Bbb{R}^n\times\Bbb{R}\to\Bbb{R}$ as follows $$ h(\mathbf{a},b)=\...
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0answers
29 views

Compare $\int_{-\infty}^{+\infty} f(x) dx \quad\text{ and }\quad \lim_{t\rightarrow +\infty} \int_{-t}^{t} f(x) dx$

I would like to compare these two integrals $$\int_{-\infty}^{+\infty} f(x) dx \quad\text{ and }\quad \lim_{t\rightarrow +\infty} \int_{-t}^{t} f(x) dx$$ My Thoughts: Let $\displaystyle I=\...
2
votes
4answers
82 views

Integral how to choose the good substitution?

I have a problem with this integral. I don't know what kind of substitution to use. $$\int t\sqrt{1+9t^4}dt$$
1
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1answer
53 views

How do I evaluate the following double integral over a general region

How do I evaluate the following integral $$ \int_{-1}^1 \int_{\arccos(y)}^{\pi} \sin(x)\sqrt {1+\sin^2 x} \,dx\,dy $$ I attempted to change the limits for the integrals $$ \int_0^\pi \int_{\cos(x)}^...
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1answer
82 views

What is the primitive of $f(x) = (x^a + b)^\frac{1}{a}$?

In studying a physical problem I was stopped by an integral that can be written in this clean way $$ \int_0^K (x^a + b)^\frac{1}{a} dx \qquad a,b \in \mathbb{R} \qquad x,K \in \mathbb{R}^+ $$ I tried ...
2
votes
4answers
111 views

what's $\int \sin (e^x)\ dx$ ??

I was working on a physics problem and I faced this integral: $$\int \sin(e^x) \, dx =$$ I tried to solve it but I could not.
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0answers
14 views

Replicate Matlab integrator block in MS excel

I created a simple diagram to solve ordinary differential equation as shown below. Simple ODE I was trying to compute the result of xf_dot manually in Ms Excel but I did not get the same answer ...
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0answers
47 views

Definite integral of product of Algebraic, Bessel's function, and exponential involving algebraic function from limits $x = u$ to $x = \infty$

I'm deriving Cumulative Distribution function (CDF) of a random variable $\Gamma$ as given below: \begin{equation} \Gamma = \frac{XY}{aX+bY+cZ+d} \end{equation} After lots of manipulation I'm stuck ...
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2answers
44 views

solution of an improper integral.

I was solving following improper integral: $$ \int\limits_0^\frac{\pi}{2}\frac{log~x}{x^a}dx $$ where $a<1$. My attempt: $0$ is the only point of discontinuity. So, $\frac{log~x}{x^a}\leq \...
4
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2answers
85 views

Computing the Integral $\int r^2 \text{J}_0(\alpha r) \text{I}_1(\beta r)\text{d}r$

I encountered the following integral in a physical problem $$I=\int r^2 \text{J}_0(\alpha r) \text{I}_1(\beta r)\text{d}r$$ where $\text{J}_0$ is the Bessel function of first kind of order $0$ and ...
2
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1answer
66 views

How to solve $y' + y^2 - 2y\sin x + \sin^2x = \cos x$

How to solve the following equation? $$y' + y^2 - 2y\sin x + \sin^2x = \cos x$$ It is necessary to determine the type and total solution. Help me please.
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1answer
21 views

Fourier transform , of a function computation

I need help with this Fourier transform computation. $$F(w)=\int_{-\infty}^{\infty} e^{-|x|+ix}e^{-iwx} \, dx$$ Need help to compute.
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0answers
127 views

Calculation of $\int_{0}^1 \frac{\sin(\ln^4(1-x))}{x}~dx$

$$I=\int_0^1 \frac{\sin(\ln^4 (1-x))}{x}dx$$ What is the closed-form evaluation of this integral? I honestly do not have a single clue how to solve this. (There is no application, but it is out of ...
1
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2answers
58 views

If $f$ is positive, decreasing, and continuous on $[0,+\infty)$ show that $\int_0^\infty f(t)dt$ converges iff $\sum_{n=0}^\infty f(n)$ converges

My Work: Integrals take infinitely small steps to get from one term to the next, whereas in a series, the distance between each term must be some tangible value ($|x_n-x_{n-1}|\ge \epsilon$). Thus, ...
1
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1answer
35 views

Solve for a variable that is within an integral and the upper bound of the integral

So I took a Calculas test today and this is a problem that I know I did not get correct. I've been thinking for some time on it and this is as far as I got, any corrections or an answer would be great,...
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0answers
36 views

Fourier transform

I am having a difficulty understanding how to approach this problem. I have state vector $$f(x)=e^{-|x|+ix}$$ and observable $$P=-id\dfrac dx $$ I want to find fourier transform of $f$? and the ...
1
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1answer
46 views

Fourier transforms having compact support

As we know, the fourier transform is a map $\mathcal{F}:L^1\rightarrow C_0$ (all with domain $\mathbb{R}$). Can one characterize the space of $f\in L^1$ such that $\mathcal{F}$ has compact support, i....
2
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1answer
37 views

Link between fundamental theorem of calculus and integral with parameters

My problem is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. We know that $$F:x\mapsto\int_0^xf(t)dt$$ is the primitive of $f$ that vanishes in $0.$ But i can rewrite $F$ ...
0
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3answers
64 views

How to evaluate integral with radian limits?

$$\int_{\pi/3}^{\pi/6} \csc \theta \cot \theta\, d\theta $$
0
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2answers
61 views

What is the interpretation of the fundamental theorem of line integrals?

The fundamental theorem of line integrals is: $$\int_{a}^{b} \nabla F \cdot \vec{dr} = F(r(b)) - F(r(a))$$ for some curve traced by $r$. What is the intuition for why this is true? The proof is ...
0
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1answer
40 views

Finding an Integral Value given other Integral Values?

I am studying for an upcoming Calculus exam, and was hoping someone here could explain how to do the following. \begin{align} \int_1^9 f(x)\,\mathrm dx&=-1\\ \int_7^9 f(x)\,\mathrm dx&=5\\ \...
0
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1answer
61 views

Evaluate $\int^e_1\frac{xe^x+1}{x}f(e^x+\ln x)\,dx$, where $g'(x)=f(x)$.

Suppose $g'(x)=f(x)$ for all $x$. Evaluate $$\int^e_1\frac{xe^x+1}{x}f(e^x+\ln x)\,dx$$ Also, what are these type of questions called?
0
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1answer
18 views

kinetic energy interation

I was reading the following article http://www.askamathematician.com/2015/03/q-why-does-kinetic-energy-increase-as-velocity-squared/ I don't understand the math of the explanation at the final of the ...