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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9 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
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2answers
22 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 ...
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0answers
11 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 ...
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1answer
37 views

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

Calculate $\int_0^1f(x)dx$,where $$\ f(x) = \left\{ \begin{array}{l l} 0 & \quad \text{if $x=0$ }\\ n & \quad \text{if $x\in(\frac{1}{n+1},\frac{1}{n}]$} \end{array} \right.$$ ...
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1answer
20 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 $$\frac12\int\left(\frac{\cos Q}{\sin Q} + \frac{\sin Q}{\cos Q}\right)\,\mathrm{d}Q$$ ...
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0answers
18 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 ...
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1answer
22 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), ...
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41 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 ...
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1answer
28 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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11 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 ...
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1answer
36 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 ...
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0answers
20 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 $$U(f,P) - L(f,P) = \sum_{i=1}^{n}\left[ f(x_i) - f(x_{i-1})\right](x_i - x_{i-1})$$ ...
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0answers
13 views

I want to compute the Fourier Transform of $g(x) = (1+e^{a x})^{1/a} \mathbf{1}_{x<0} e^{-x}$

I would like to compute the fourier transform of $g(x) = (1+e^{a x})^{1/a} \mathbf{1}_{x<0}\ e^{-x}$ Question 1: Does the transform exist in any sense?. The sufficient condition ...
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1answer
7 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 ...
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3answers
54 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 ...
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1answer
20 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 ...
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1answer
45 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 ...
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2answers
107 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 ...
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0answers
54 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,\ \ ...
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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 $f^{-1}$ ...
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3answers
69 views

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

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.
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4answers
110 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 $$\int_0^1\frac{\arctan^2 x\ln x}{x}\,dx?$$
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3answers
73 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 ...
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1answer
35 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.
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14 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 $y=0$ ...
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1answer
13 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 ...
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46 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, ...
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5answers
620 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 ...
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3answers
75 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 ...
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1answer
32 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} ...
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0answers
33 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 ...
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1answer
50 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 ...
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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 ...
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3answers
67 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 ...
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1answer
25 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 ...
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1answer
33 views

Proving integration formulas from scratch

Prove the following integration formulas from scratch? (I uploaded them)
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1answer
42 views

Spectral Measures: Riemann-Lebesgue

Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. Consider a Borel spectral measure $E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$. Denote its associated ...
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1answer
38 views

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

How to calculate "general" integral: $\int\limits_{a}^{b}f(x)^2dx$?
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4answers
122 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 ...
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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 ...
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3answers
57 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?
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2answers
143 views

Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$

At first sight it looks like the integral below $$\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$$ can be evaluated by using some geometric series. What else can we do? Is there a fast easy ...
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0answers
12 views

Solving ODE involving matrices

We have a given ODE $ K(x)_{_{3 \times 3}}=xC_1K(x)+x^3C_2K'(x) \tag 1$ where $C_1,C_2$ are constant skew symmetric matrices of dimension $3 \times 3$ with determinant $0$. How do we solve ...
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1answer
16 views

how to evaluate $\int \left(\hat{V}\times \frac{d^2\hat{V}}{dt^2}\right)dt$?

if $\hat{V}\left(t\right)$ is a vector function of $t$, find the indefinite integral $\int \left(\hat{V}\times \frac{d^2\hat{V}}{dt^2}\right)dt$ To solve thi first i find for the integrand with ...
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0answers
20 views

CDF of ratio of Gamma distribution with different parameters

Let $X$ be gamma distributed random variable with parameters $a$ and $b$. Let $W$ be gamma distributed random variable with parameters $c$ and $d$, such that \begin{equation} f_X(x) = ...
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4answers
38 views

solve indefinite integral

I have this indefinite integral $\int 3 \sqrt{x}\,dx$ to solve. My attempt: $$\int 3 \sqrt{x}\,dx = 3 \cdot \frac {x^{\frac {1}{2} + \frac {2}{2}}}{\frac {1}{2} + \frac {2}{2}}$$ $$\int 3 ...
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0answers
24 views

Understanding integration and substitution

I'm an undergraduate student in EE. I often see that when talking about voltages, curents ... being expressed like functions of some independed variable (time) and when calculating integrals people ...
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2answers
30 views

Integral involving radicals

Can anyone give some hint as to how to proceed in solving this integral: $$ \int \frac{u}{\sqrt{R^{2}+r^{2}-2Rru}}\mathrm{d}u $$
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1answer
19 views

Limit integration

I have: $$ F(a)=\int_0^a(x^2+1)e^{-x/2} dx $$ I have done the integration: $$ \int(x^2+1)e^{-x/2}=-2e^{-x/2}(x^2+4x+9)+C$$ What is (if possible): $$ lim_{a \to \infty} F(a)$$ I tried: $$ ...
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0answers
22 views

Jacobian determinant of unitary transformation

Is the Jacobian determinant of a unitary transformation equal to one? I ask because I get that impression from the appendix of this paper. They have spherical coordinates for two particles, ...