Questions tagged [integration]

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 the integrals being considered. This tag often goes along with the (calculus) tag.

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2answers
30 views

Order of factors in partial decomposition

Is there a protocol for deciding which denominator fraction goes under A and which goes under B during partial decomposition? Doing this question: integral $(5x-5)/(3x^2-8x-3)$ I factored the ...
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0answers
26 views

Integration by parts with sign(X) function

I have the following equation: and I assume we use integration by parts to obtain the given answer: I am not sure what to chose as u or v. The integral of sign gives me 0 and the derivative is ...
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0answers
68 views

A generalization involving a logarithmic integral

In the preprint, A note presenting the generalization of a special logarithmic integral by Cornel Ioan Valean, it is given the following generalization, Let $n\ge1$ be a positive integer. Then, \...
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0answers
73 views

Cubes under the graph of $x^2$

Let $D$ be the set under the graph of $f(x)=x^2$ for $0\leq x\leq 1$, that is, $D= \{ (x_1,x_2) : 0\leq x_1\leq 1, 0\leq x_2\leq x_1^2 \}$. Denote by $A_{r}(y)$ the $2$-dimensional cube of side length ...
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1answer
72 views

Compute, $I(x)=\int_{-1}^{1}\frac {e^{-x\mu}}{\sqrt{1+\mu^2}}~ \mathrm d\mu.$ [closed]

Is there a closed form of the following integral $$I(x)=\int_{-1}^{1}\frac {e^{-x\mu}}{\sqrt{1+\mu^2}}~ \mathrm d\mu$$
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1answer
223 views

Integrating 2-form and change of variables question

There's a nice quote online that says: "The whole point of differential forms is that integration of forms is designed to work consistently with the change of variables formula. Or another way ...
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0answers
65 views
+50

Integral over Markov Process

I have two integrals $$\int_{0}^{\infty} \mathbb{E}[\cos 2 \psi(0) \cos 2 \psi(z)] d z$$ $$\int_{0}^{\infty} \mathbb{E}[\sin 2 \psi(0) \sin 2 \psi(z)] d z$$ where $\psi(z)$ is randomly varying ...
3
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1answer
52 views

Minimizing Functional on L2 space

Let $X$ be a non-empty Borel subset of $\mathbb{R}$ and consider the finite-measure space $(X,\mathcal{B}(X),\mu)$. Fix $y,f^1,\dots,f^n \in L^2_{\mu}(X)$ and define the objective function $$ \begin{...
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2answers
61 views

For which $~p,~q~$ does $\int_2^∞ x^p⋅(\ln x)^q dx$ converges? [duplicate]

For which $~p,~q~$ does $$\int_2^∞ x^p⋅(\ln x)^q dx$$ converges? I have no idea how to start. What should I do?? Edit : Thanks for help!!
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2answers
52 views

How to change the limits of given integral

I need to evaluate the following integral : $$\int_{ -1}^1\int_{1+x}^1\cos\left(x+y\right)e^{(y-x)}dydx$$ I know that I need to change the variables by using substitution $u = x+y$ and $v =y-x$ but ...
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1answer
45 views

Integrating a product of Heaviside step functions

I would like to deal with the following integral: $$ f(\vec y) = \int\limits_{\mathbb R^3}\text{d}^3\vec x\, H(a-|\vec x|)\,H(b-|\vec x+\vec y|) \quad a,b>0 $$ where $H(x)$ is the Heaviside step ...
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1answer
65 views

Derivative of $\int_0^tf(x,t)dx$?

Derivative of $\int_0^tf(x,t)dx$? Fundamental theorem of calculus works for $\int_0^tf(x)dx$ $$\frac{d}{dt}\int_0^tf(x)dx=f(t)$$ but how about this case?
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1answer
59 views

What is $\int_0^{p} e^{\frac{-(\ln(x)-\mu)^2}{2 \sigma^2}}\, dx$

Is there a closed form solution for this definite integral? I'm trying to derive the cumulative distribution function for a statistical distribution in the lognormal family. $$\int_0^{p} \exp\left({\...
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2answers
60 views

The convolution (?) of $a/ (x^2 + a^2)$

Let $a$ be a nonzero real number and $$ f(x) = \frac{1}{\pi} \frac{a}{x^2 + a^2}.$$ Then compute $$\int_{-\infty}^\infty f(x) dx$$ and, for $t$ a real number, $$\int_{-\infty}^\infty f(x)f(...
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2answers
95 views

Show that $\sum\limits_{k=1}^{+\infty}[\frac{1}{k}-\frac{1}{k+t}]$ is uniformely-convergent

Studying notes about Gamma function,i came across the following problem that i was unable to solve. Let's be $\phi = \frac{d}{dx}(log(F(x))),$ where $F(x) = \int_{0}^{+\infty} t^{x}e^{-t}dt$, ...
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1answer
22 views

How to sketch these bounds

I need to calculate a double integral over a region as follows: $$ 0 ≤ x − y ≤ 1 $$ $$2 ≤ x + y ≤ 3 $$ But I'm not sure what this region looks like, I'm having some trouble understanding in ...
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3answers
115 views

Find the increasing function $f:\mathbb{R} \to \mathbb{R}$ such that $\int_{0}^{x}f(t)\,dt=x^2$ for all $x\in \mathbb{R}$

Find the increasing function $f:\mathbb{R} \to \mathbb{R}$ such that $$\int_{0}^{x}f(t)\,dt=x^2\text{ for all }x\in \mathbb{R}$$ My solution: Let $F:\mathbb{R} \to \mathbb{R}$,$F(x)=\int_{0}^{x}f(t)\,...
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0answers
22 views

Irrotational/conservative vector field on the plane

I just would like to check if the solution provided for the following exercise is correct (I just have some doubts about the existence of the partial derivatives I computed in order to prove that the ...
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0answers
36 views

Solving a Probabilistic Integral Equation

The following problem arose for an example that I wanted to give in my paper: Prove (or disprove) that there exists a function $g\colon {\mathbb R_+} \to \mathbb{R}$ such that $$\displaystyle \int_{\...
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0answers
58 views

Integral limit that behaves like a delta function

I am having difficulty seeing why this integral reduces to $1$ or $0$, just like the delta function. In the following statement, $\Delta$ is small, whereas $R$ goes to infinity: $$\lim_{R\rightarrow \...
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2answers
49 views

Why isn't this integral equal to $0?$

I am kinda stuck in this quite simple integral and I can't understand what I'm doing wrong. According to one of my uni's tests, the closed-path integral of $$F(x,y)=\Big(\frac{y}{2x^2+y^2}, \frac{-x}{...
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1answer
62 views

What is the integral for this function? [closed]

How can I integrate the following function and what is the resulting integrated function? $$\int_0^{\infty} e^{-t \cdot \lambda_S} \cdot \sum_{x=k}^{n} \binom{n}{x} \cdot e^{-t \cdot \lambda_P \cdot ...
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2answers
22 views

Evaluation of an expectation involving a product of normally distributed rvs

This should be not too hard but I am stuck on this for 2 hours now... I have a normally distributed rv with mean 0 and variance 1 and I want to calculate $$E[X\cdot(1_{X\in A}X - 1_{X\in\complement A}...
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0answers
33 views

Upper bound for the multiple integral

I know that the multiple integral $$\int_{R^d} (1+|\omega|^{2k})^{-1}d\omega$$ is convergent for 2k>d, but can anyone find a closed form solution, or otherwise provide a pretty tight upper bound (in ...
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0answers
29 views

Volume between a hypersurface and a height function

If I have a smooth $(n-1)$-dimensional hypersurface $N$ in $R^n$ and a smooth positive height function $f:N \to R$ bounded with $f=0$ on $\partial N$, how can I compute the volume enclosed by $N$ and ...
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2answers
77 views

Evaluating $\int_0^\infty x\cdot e^{-n^2x-\frac{\pi^2}{x}(k^2+j^2)}\,dx$

How could the integral: $$\int_0^\infty x\cdot e^{-n^2x-\frac{\pi^2}{x}(k^2+j^2)}\, dx$$ be evaluated? I know the similar looking integral: $$\int_0^\infty \frac{e^{-n^2x-\frac{\pi^2}{4x}(k^2+j^2)}}{\...
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0answers
53 views

What's wrong with this proof that the integral is the area under the curve?

There is a purported proof in this YouTube video, but I know that the real proof that the integral is the area under the curve is much more complicated, and involves first proving it for step ...
4
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3answers
199 views

How to compute $\int_0^\pi {\sin x\sin nx\over 1-2a\cos x+a^2} dx$

$$\int_0^\pi {\sin x\sin nx\over 1-2a\cos x+a^2} dx$$ With some bit of tinkering with Desmos, I've got to know that the answer is ${\pi \over 2} a^{n-1}$. But can you help prove that? Sorry for the ...
4
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3answers
94 views

Integration of $ \int x^{2} \sqrt{2x-6} dx $

$$ \int x^{2} \sqrt{2x-6} dx = ?$$ My Attempt: by partial integration $$ \int x^{2} \sqrt{2x-6} dx = \frac{x^{2} (2x-6)^{3/2}}{3}- \frac{2}{3} \int x(2x-6)^{3/2}dx$$ continuing partial integration $$...
1
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2answers
50 views

Evaluate $ \int \frac{3x^{2} (\csc{\sqrt{x^{3}-6}})^{2}}{\sqrt{x^{3}-6}} dx $

$$ \int \frac{3x^{2} (\csc{\sqrt{x^{3}-6}})^{2}}{\sqrt{x^{3}-6}} dx = ? $$ Attempt: $$ \int \frac{3x^{2} (\csc{\sqrt{x^{3}-6}})^{2}}{\sqrt{x^{3}-6}} dx = \int \frac{3x^{2}}{\sqrt{x^{3}-6} \sin^{2}\...
4
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2answers
64 views

How to express integration $k$ times in a row?

Differentiation can be expressed $k$ times in a row as either $f^{(k)}(x)$ or as $\frac{d^n}{dx^n}f(x)$. How do I express indefinite integration $k$ times in a row?
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79 views

Do I have to use integration to solve this exercise?

everybody, I am doing an exercise but I am not quite sure if the way I am going to resolve is the right one. Can I have your advice on it, please? Exercise: Thanks to the new acquisition of advanced ...
1
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1answer
65 views

How does one derive the following formula of integration?

$$\int_0^\infty\frac{\exp{\left(-\frac {y^2}{4w}-t^2w\right)}}{\sqrt {\pi w}}dw=\frac{\exp(-ty)}t$$ for $t$ and $y$ positive. This integral is useful in the following context: suppose we are given $$\...
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0answers
53 views

Eulerian type of integral

Could anyone give me some light on how to evaluate $$ \int_{0}^{1} \frac{\sqrt{1-t}\sqrt{a+t}}{t^2}\left(1-e^{-t}-2te^{-t} \right)dt$$ where $a$ is a positive real constant. I tried several ...
1
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2answers
80 views

Does Tanh(x) have areas which are concave up or down?

I know the point of inflection is at x = 0, however I am struggling with the second derivative test to identify the places where Tanh(x) is concave up or down.
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1answer
49 views

Get around Exponential Integral in indefinite integral calculation?

I have two functions, $f(x) = (a_0*e^{(-a_6*(x + (a_1*x + a_2)))})$ and $g(x) = abs(a_3*x^2 + a_4*x + a_5)$ I'm trying to find the integral $\int{f(x)/g(x) dx}$. Wolfram gives back : $(a_0 exp(-1/2 ...
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1answer
30 views

Finding the area under a curve when the area is bounded by 3 curves.

The following problem is from the book, Calculus and Analytical Geometer by Thomas and Finney. Problem: Find the area of the region bounded by the given curves and lines. $$ y = x, y = \frac{1}{ \sqrt{...
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1answer
41 views

Validity of interchanging limit and integral under non-uniform convergence!!

Can I interchange the limit and integral of a sequence of functions which is not uniformly convergent in $[0,1]$ i.e $f_n \not\to f$ uniformly is it true that $\int_0^{x_n}f_n \to \int_0^1f$ for $x_n\...
4
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2answers
65 views

Evaluating the integral: $ I = \int e^{\frac xa} \sin x \, \mathrm dx$

Evaluating the integral: $$ I = \int e^{\frac xa} \sin x \, \mathrm dx \tag {1}$$ This question was asked in CBSE Board 12th Grade (India). So, here was the approach I made. Proposition 1: $$ ...
1
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1answer
58 views

Verification: If $\int_a^bf(x)dx=0$ then there is $x_0\in [a,b]$ for which $f(x_0)=0$

Old exam question: Let $a,b\in\mathbb{R} : a<b$ and let $f$ be a continuous function, $[a,b]\rightarrow\mathbb{R}$. Show that if $$\int_a^bf(x)dx=0$$ then there is $x_0\in [a,b]$ such that $f(...
0
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1answer
33 views

Choice of partition for Riemann integrationn

Let $f:[0,1]\rightarrow \mathbb{R}$ be a function. If $P=\{0=a_0 < a_1 < \cdots < a_n=1\}$ is any partition of $[0,1]$, then we define $U(p,f)$ and $(L(p,f)$, the upper and lower Riemann sums....
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3answers
58 views

Calculating $\int_{0}^{\sqrt{3}} \frac{dt}{\sqrt{1+t^2}}$.

Calculate $\int_{0}^{\sqrt{3}} \frac{dt}{\sqrt{1+t^2}}$. So we let $t = \tan{(x)}$ so $1+t^2 = 1+\tan^2{(x)} = \sec^2{(x)}$ which means $$\int \frac{dt}{\sqrt{1+t^2}} = \int \frac{1}{\sec^2{(x)}}\...
1
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1answer
43 views

Fourier sine expansion for $x(\pi - x)$

I'm trying to find the Fourier sine expansion for the function $f(x)$ = $x(\pi - x)$ for the interval $0 \leq x \leq \pi$. I think I am supposed to find $\sum_{k=1}^{n} b_k \sin{(kx)}$ where $b_k = \...
10
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1answer
192 views

Is there a closed form for $\int_a^b\frac{{\rm arccosh}x}{\sqrt{(x-a)(b-x)}}$?

I have solved $$\int^3_1\dfrac{{\rm arccosh}x}{\sqrt{(x-1)(3-x)}}{\rm d}x=4G$$where $G$ is Catalan constant, by rewriting it into double integral $$\iint_{[0,\pi]\times[0,\pi]}\ln(2-\cos x-\cos y){\rm ...
2
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3answers
60 views

Find integral area with respect to x

The question is finding the area enclosed by the curves x=$y^2$ and x+2y=8 using both x and y integrals Graph for reference Purple is x+2y=8, red is x=$y^2$ First I found the limits by letting x=8-...
1
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2answers
39 views

Computing $\int\frac {du}{\sqrt{u^2 + s^2}} =\log \lvert (u + \sqrt{u^2 + s^2}) \rvert$ with a substitution

Can someone please show me where I am going wrong? It seems there is a contradiction in the formula for the the following integral. $$\int\frac {du}{\sqrt{u^2 + s^2}} = \log \bigl\lvert u + \sqrt{u^...
4
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2answers
122 views

Calculus - indefinite integration

The integral in which I am interested in is $$\int x(x^3+1)^{33}\mathrm{d}x$$ I tried to solve by substituting $x^2 = t$, but it didn't help. I find a solution by expanding it with the help of ...
3
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2answers
132 views

There is $f$ such that $\int_0^1 f dx = \lim_{c \downarrow 0} \int_c^1 fdx$, but for $|f|$ this limit does not exist. How is that possible?

If $f$ in integrable on some interval $[a,b]$ then we know that $\lvert f \rvert $ is also integrable on that same interval. There is a problem in Rudin's Principles of Mathematical analysis such ...
1
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0answers
29 views

Integration by partial fractions: repeated denominator factor

It is stated, that given a factorization $$g(x) = (x - a)^m(bx^2 + cx + d)^n$$ we can create partial fractions of the type: $$\int f(x)/g(x) dx = \int \sum_{i = 0}^m \frac{A_i}{(x - a)^i} + \sum_{j =...
1
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1answer
23 views

Apostol equation for partial fraction simplification

In Apostol, the case that the partial fraction is presented as $\frac{C}{(u^2 + a^2)^m}$ can be done with the following reduction formula: $$\int \frac{du}{(u^2 + a^2)^m} = \frac{1}{2a^2(m-1)} \frac{...