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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19 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
0
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1answer
33 views

Evluating triple integrals via Spherical coordinates

Use Spherical coordinates to evaluate the triple integral $$\iiint_{\mathrm{x^2+y^2+z^2<z}}\sqrt{x^{2}+y^{2}+z^{2}}\, dV,$$ What I tried Converting $x^2+y^2+z^2<z$ to Spherical coordinates ...
4
votes
0answers
417 views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
-5
votes
1answer
39 views

How to calculate this kind of integral? [on hold]

General form: $$ \int \sqrt{\alpha x^3 + \beta x^2 + \gamma x + \delta} \, dx $$ Example: $$ \int \sqrt{\frac{2}{3} x^3 + x^2 + 4} \, dx $$ Please describe more details.(I'm a freshman.) I will ...
6
votes
0answers
67 views

Real analytic methods for the following integral [duplicate]

A few days back, the following integral was posted $$\int_0^1 x^x(1-x)^{1-x}\sin(\pi x)\,dx=\frac{\pi e}{24}$$ The integral was answered using complex analysis tools but I am interested in other ...
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2answers
25 views

feedback on my solution (improper integral)

i have done this improper integral but i am not sure if i have followed the correct procedure or my answer is correct. Please help!
3
votes
1answer
29 views

feedback on my solution (integration)

I need help in this problem. I managed to find the answer for this problem by using mathmatica but cannot do the working for it. i have done most of it but i am stuck on the last part.
-2
votes
0answers
32 views

how to calculate the following definate integral [on hold]

∫f(a,b,x)dx= Antiderivative or integral could not be found.
0
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0answers
41 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Proof:$\forall a,b;c,d\in\mathbb{R},a<b,c<d.$ $f (x+y) $ is ...
2
votes
1answer
56 views

Really tricky integration-----double U and trig substitution

The definite integral $h(x) = \sin x/(1 + x^2)$ on the closed interval $[-1,1]$ $\tan^2(x) + 1 = \sec^2(x)$, $x = \tan(@)$ $$x = \tan(@)$$ $$dx = \sec^2(@) d@$$ now I have to find sin9x0 in terms ...
0
votes
0answers
12 views

integration featuring the unit step function

Compute the following integrals I don't know how to use MathJaX so here's a link to the image of the integrals where u(t) is the unit step function and σ is some variable of integration
1
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2answers
45 views

How do I approach this double integral?

Let $R$ be the region inside $$x^2+y^2 = 1$$ but outside $$x^2+y^2 = 2y$$ with $x \ge 0 $ and $y \ge 0$ Let $$u = x^2 + y^2$$ and $$v = x^2+ y^2 - 2y$$ Compute $ \iint_R xe^y dxdy$ using this change ...
0
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0answers
17 views

Integration by parts partial derivatives

Given $$\int_x \int_t \Big( \frac{\partial}{\partial t}u(x,t) + \frac{\partial}{\partial x}f(u(x,t)) \Big) \phi(x,t)~~ dt dx = 0$$ How can I apply integration by parts in order to have the ...
1
vote
1answer
45 views

Calculate $\int_0^1f(x)dx$

Calculate $\int_0^1f(x)dx$,where $$\ f(x) = \left\{ \begin{array}{l l} 0 & \quad \text{if $x=0$ }\\ n & \quad \text{if $x\in(\frac{1}{n+1},\frac{1}{n}]$} \end{array} \right.$$ ...
0
votes
1answer
24 views

Integrate $\int \csc 2Q\,\mathrm{d}Q$

I need to use $\cot Q+\tan Q=2\csc 2Q$ to integrate $$\int \csc 2Q\,\mathrm{d}Q.$$ the integral becomes $$\frac12\int\left(\frac{\cos Q}{\sin Q} + \frac{\sin Q}{\cos Q}\right)\,\mathrm{d}Q$$ ...
1
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2answers
55 views

Integration of $F(\sum_k x_k)$ over positive orthant

Problem Suppose we some function $F\left(\sum\limits_{k=1}^n x_k\right)$ over the positive orthant $[0,\infty)^n$. Show that this this is proportional to the integral $\int\limits_0^\infty ...
0
votes
1answer
27 views

Is there a clever way to determine negative area of an integral?

Given some continuous, integratable function f(x) that has only positive area over a range from x1 to x2...is there a way to determine the negative area of the integral of f(x) - c (from x1 to x2), ...
8
votes
1answer
104 views

Closed-form of $\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx$

Inspired by this question, is there a closed-form of $$\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx\,?$$ Here $n \in \mathbb{N_+}$. In the answers to the question above we could find proofs of ...
0
votes
1answer
30 views

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$ I integrated it once to get, $2x + \sin x + C$, $C$ being a constant. Then I integrated it a second time to get $x^2 - \cos x ...
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votes
0answers
13 views

ODE - Laplace transform

I have an ODE $\psi^{'}(s)_{3 \times 3}=(A+Bs)_{3 \times 3}\psi(s)_{3 \times 3} \tag1$ where A,B are constant skew symmetric matrices with zero determinant. $\psi(s)$ is a rotation matrix. It implies ...
2
votes
1answer
47 views

Compute a multiple integral$\iint_{[0,1]^2} (xy)^{xy} dxdy$

$$\text{Compute} :\iint_{[0,1]^2} (xy)^{xy} dxdy$$ I am thinking about changing the variable, $x=u,y={v \over u}$.But it doesn't work. I just found that the answer is$\int_0^1 t^t dt$.Maybe my idea ...
2
votes
0answers
21 views

If $f$ is increasing, then for all $n\in\mathbb{N}$ there exists $P_n$ : $U(f,P)-L(f,P) \leq (b-a)/n$

I've already proven that, if $f:[a,b] \to \mathbb{R}$ is continuous and increasing, with $a,b\in \mathbb{R}$, then $$U(f,P) - L(f,P) = \sum_{i=1}^{n}\left[ f(x_i) - f(x_{i-1})\right](x_i - x_{i-1})$$ ...
0
votes
1answer
13 views

Joint CDF from conditional cdf

I would like to derive an expression of the following joint CDF $P[X \leq x,Y \leq y]$ based on the conditional CDF $P[X \leq x | Y=y]$ and the pdf $P[Y=y]$ that are considered to be known. I get a ...
3
votes
3answers
57 views

How to compute $ \int e^{-st} \sin(2t) dt $

Wolfram Alpha shows me the result of $ \int e^{-st} \sin(2t) dt $ . However it doesn't let me see the step to step solution. Then I tried to do this by hand as the solution didn't look "too ...
1
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1answer
33 views

Application of the mean value theorem for Integrals

Suppose that $f(x)$ is a differentiable function in $[a,b]$, $f^{'}(x)$ is a monotone decreasing function in $(a,b)$, and $f^{'}(b)>0$. So how to prove that $$ \big \vert \int_a^b \cos ...
1
vote
1answer
53 views

How to integrate $e^{\sqrt{2x}}$?

I think this problem requires integration by substitution and integration by parts, but I seem to get stuck each time I try to solve it. And I'm not sure whether '$u$' should be equal to $\sqrt{2}$ or ...
1
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2answers
119 views

Differentiating with respect to the limit of integration

I'm confused about problems involving differentiation with respect to the limit of an integral, I just want to check that my understanding is correct. For example, are the following statements ...
1
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0answers
60 views

What is the recurrence relation between $a_n, a_{n-1}$ , $a_n = \int_0^1 {x^n}\tan\left( \frac{\pi}{4}x\right) dx$

I would appreciate if somebody could help me with the following problem Q: What is the recurrence relation between $a_n, a_{n-1}$ ? $$ a_n = \int_0^1 {x^n}\tan\left( \frac{\pi}{4}x\right) dx,\ \ ...
0
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0answers
48 views

Can this be expressed by a contour integral?

Let $f(z)$ be a real entire function of the form $f(z) = a_1 z + a_2 z^2 + ...$ such that $0 < a_{n+1} < a_n$. Consider $g(x) = f^{-1}(f(x)-f(x-1))$ where $x$ is a positive real and $f^{-1}$ ...
7
votes
3answers
82 views

Improper integral : $\int_0^{+\infty}\frac{x\sin x}{x^2+1}$ [on hold]

How to evaluate the following improper integral : $$\int_0^{+\infty}\frac{x\sin x}{x^2+1}\,dx$$ I have tried integration by parts and variable substitution, but it didn't work.
4
votes
4answers
183 views

Inverse Trigonometric Integrals

How to calculate the value of the integrals $$\int_0^1\left(\frac{\arctan x}{x}\right)^2\,dx,$$ $$\int_0^1\left(\frac{\arctan x}{x}\right)^3\,dx $$ and $$\int_0^1\frac{\arctan^2 x\ln x}{x}\,dx?$$
3
votes
3answers
85 views

Evaluate integral: $\int_0^{+\infty}\frac{\cos{bx}-\cos{ax}}{x}dx$

It seems that $\displaystyle\int_0^{+\infty}\frac{\cos x}{x}$ is divergent, so how to solve this problem? $$\int_0^\infty \frac{\cos bx -\cos ax}{x}\, dx\quad,\quad\mbox{where}\,a,b>0$$ It's ...
0
votes
1answer
36 views

Calculate integral of $\ln(z)$ using the residue theorem

Please is it possible to calculate $\int_{C(0,1)}\ln(z)\,dz$ using the residue theorem? Thank you for your help.
0
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0answers
15 views

Calculating volumes using integral.

Given $y=x,y=0,x=2$ and $x=7$. Calculate the volume6 about $x=1$. I just need to get the concept right. Please tell me what mistake I did here. The region looks like a trapezium right? From $y=0$ ...
0
votes
1answer
15 views

Integral of a normal function multiplied by heaviside and delta functions

$\int_{-\infty}^{\infty} e^{2t}u(\tau - t)t^{2}\delta(t)dt$ Hi! How would I go about computing this integral? I understand I can change one of the integration limits and eliminate the heaviside ...
8
votes
0answers
63 views

Closed form of a difficult definite integral

I'm looking for a closed-form expression for the value of this integral: $$I=\int_0^\pi \frac{\sin(x)}{\sqrt{x^3+x+1}} dx$$ The graph of the integrand looks like this: $\hskip 2.4 in$ Numerically, ...
2
votes
5answers
643 views

How can I show that these integrals are zero

How can I show that these integrals equal $0$ when $n$ and $m$ are both integers and $n \neq m$? $$\int_{-\pi}^{\pi}\sin(mx)\sin(nx)dx = \int_{-\pi}^{\pi}\cos(mx)\cos(nx)dx = 0$$ I'm able to show that ...
2
votes
3answers
86 views

If $f'(x)=f(x)+\int_{0}^{1}f(x)\,dx$ and $f(0) = 1,\,$ then what is the value of $\, \int_0^1 f(x)\,dx=$?

If $\displaystyle f'(x)=f(x)+\int_{0}^{1}f(x)\,dx\,$ and $\,f(0) = 1.$ Then what is value of $\displaystyle \int f(x)\,dx\,?$ $\bf{My\; Try.}$ Let $\displaystyle \int_{0}^{1}f(x)\,dx = A\;,$ Then ...
1
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1answer
33 views

Any Easier way to integrate:$\iint\limits_D{e^{x+y}}d\sigma,D=\{\left . (x,y) \right ||x|+|y|\leqslant1\}$

This is my way: \begin{align} \iint\limits_D{e^{x+y}}d\sigma & = \int_{-1}^0e^xdx\int_{-x-1}^{x+1}e^ydy + \int_0^1e^xdx\int_{x-1}^{-x+1}e^ydy \\ & = \cdots \\ & = e-e^{-1} ...
1
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0answers
36 views

How do I do this double integral (change of variable)

$B$ is the region bounded by $xy = 1$, $xy = 3$, $x^2 - y^2 = 1$, $x^2 - y^2 = 4$ Find $$\iint\limits_{B}x^2 + y^2 \,dx\,dy$$ using the change of variables: $$u = x^2 - y^2$$ $$v = xy$$ So I think ...
4
votes
1answer
52 views

Is there an alternative way to solve this integral?

I was given the integral $$\int \frac{2}{e^{-x}+1}dx$$ Here is my method to get the (correct) solution: $$\int \frac{2}{e^{-x}+1}dx$$ $$=2\int \frac{1}{e^{-x}+1}dx$$ $$=2\int ...
0
votes
0answers
19 views

Complex Fourier coefficients and series

I need help trying to find the complex Fourier coefficients for the functions $\cos(3x)$ $\sin(2x)$ I know the equation for finding the coefficients and how to plug it in but I'm confused in how ...
1
vote
3answers
71 views

Fallacy - where is the mistake?

Could anyone help me to find the mistake in this fallacy? Because the actual result for $I$ is $\pi/2$ \begin{equation} I = \int_{0}^{\pi} \cos^{2} x \; \textrm{d}x \end{equation} \begin{equation} I ...
1
vote
1answer
26 views

Solution to Differential Equation $\left( 1-2\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)+2 \lambda h(x,y,z)=0$

I'm trying to solve the following Differential Equation: $\left( 1-2\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)+2 \lambda h(x,y,z)=0$ The unknown function is $w(x,y,z)$. The ...
0
votes
1answer
37 views

Proving integration formulas from scratch

Prove the following integration formulas from scratch? (I uploaded them)
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votes
2answers
51 views

Spectral Measures: Riemann-Lebesgue

Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. Consider a selfadjoint Hamiltonian $H:\mathcal{D}\to\mathcal{H}$. Denote its associated Borel spectral measure by: ...
0
votes
1answer
39 views

How to calculate “general” integral $\int\limits_{a}^{b}f(x)^2dx$?

How to calculate "general" integral: $\int\limits_{a}^{b}f(x)^2dx$?
11
votes
5answers
200 views

Integral: $\int_0^{\pi/12} \ln(\tan x)\,dx$

I am trying to evaluate: $$\int_0^{\pi/12} \ln(\tan x)\,dx$$ I think the integral is quite simple but I am having a hard time evaluating it. I started with the result: $$\int_0^{\pi/4} \ln(\tan ...
0
votes
1answer
37 views

What is this integration “method” name?

I see that people often write this equality: $$\int\limits_a^bf(x)\,\mathrm dx=\int\limits_{f(a)}^{f(b)}f(x)\,\mathrm df(x)$$ when dealing with functins in general, that is when something is trying ...
1
vote
3answers
62 views

The value of $\int_0^{2\pi}\cos^{2n}(x)$ and its limit as $n\to\infty$

Calculate $I_{n}=\int\limits_{0}^{2\pi} \cos^{2n}(x)\,{\rm d}x$ and show that $\lim_{n\rightarrow \infty} I_{n}=0$ Should I separate $\cos^{2n}$ or I should try express it in Fourier series?