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 ...

learn more… | top users | synonyms (3)

26
votes
3answers
618 views

Integral of Combination Log and Inverse Trig Function

Does the following integral have a closed-form ?: \begin{equation} \int_{0}^{1}{\ln\left(\,x\,\right) \over 1 + x}\,\arccos\left(\,x\,\right) \,{\rm d}x \end{equation} This integral has been ...
4
votes
4answers
161 views

Integral $\int_{1}^{\infty} \frac{\log^3 x}{x(x-1)} dx$

How do I arrive at the closed form expression of the integral $$\displaystyle\int_{1}^{\infty} \dfrac{\log^3 x}{x(x-1)}dx$$ Most probably the closed form is $\dfrac{\pi^4}{15}$
0
votes
0answers
29 views

Integral of Hypergeometric Function with polynomial, power, exponential and logarithm function

I am trying to work out this integral. If there is no closed form, can you think of any approximations to it?
0
votes
3answers
73 views

General closed form of $\int_{0}^{\theta_o} \dfrac{1}{\sqrt{\cos \theta-\cos \theta_o}} d\theta$

I once asked a question about how to integrate the reciprocal of the square root of cosine. Is there a general closed form for the integral $$\int_{0}^{\theta_o} \dfrac{1}{\sqrt{\cos \theta-\cos ...
0
votes
1answer
27 views

Associating a function $v \in H^{k+1}(K)$ with a polynomial function with equal integrals of derivatives

I read the following: For all $v\in H^{k+1}(K)$ we can associate a polynomial $p\in P_k$ (space of polynomial functions with degree $\leq k$), defined by $\forall \alpha \in N^n$, with ...
0
votes
5answers
71 views

Find $\int\sqrt{(x-2)/(x-1)}\,dx$

I am having difficulty with these type of problems. Can anyone explain how to approach the problems of the form $\int \sqrt{\dfrac{x-a}{x-b}}\hspace{1mm}dx$
0
votes
1answer
50 views

Gamma and a poisson distribution

All I know is that $P(X=x)=e^{-5x}\frac{5^x}{x!}$ where $x\geq 0$. The formula that im using is $E(Y)=\int E(Y|X=x)f_x(x) dx $ where $f_x(s)\int f(x,y) dy $=. Also I guess that from the hint that ...
0
votes
2answers
32 views

Two very similar solutions to a differential equation through two different methods

In our differential equation class, we learned of two methods to solve elementary differential equations: integration factors and seperation. We had to solve the differential equation (k is a ...
7
votes
4answers
182 views

Calculus Question: Improper integral $\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx$

How to evaluate integral $$\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx?$$ I tried substitution $x=u^3$ and I got $3\displaystyle\int_{0}^{\infty}u \cos(2u^3+1)du$. After that I tried to use ...
3
votes
0answers
58 views

An integral with a decaying exponential with rational exponent

I was working on some mathematical derivations while I faced this integral: $$\Large \int_0^\infty x^{\alpha-1}e^{-\beta x} e^{-\lambda \left[\frac{x^2}{2x+\eta}\right]}\ \mathrm{d}x \quad .$$ Does ...
5
votes
3answers
141 views

Find $\int \sinh^{-1}x\hspace{1mm}dx$

Find $\int \sinh^{-1}x\hspace{1mm}dx$ $ $ I am asked to use the following Equation: $$\int \tan^{-1}x\hspace{1mm}dx= x\tan^{-1}x-\ln(\sec(\tan^{-1}x))+C$$ $ $ The confusing part is : What has ...
0
votes
0answers
49 views

What is the Riemann surface of the exponential integral?

I have recently encountered a differential equation whose solution has a term \begin{equation} \frac{1}{2}e^{-\frac{1}{2 \varepsilon} e^{i \tau}} \int_{\tau_0}^\tau e^{\frac{1}{2 \varepsilon} e^{i ...
1
vote
2answers
55 views

Integrating Trig Functions

How you I solve the following integral? $\int_{-\pi}^\pi \cos(5x)\cos(nx)dx$ I know I need to use the ${1\over 2}(\cos(u-v)+\cos(u+v))$ but I keep getting zero.
1
vote
0answers
29 views

a question about integration by parts

Suppose that $t f(t) \to 0$ when $t \to \infty$ and $t f(t)\to 0$ when $t \to 0$. For the following integral, $$I(z)=\int_0^{\infty} f(t) \cos (z t) \mathrm{d}t,\qquad z>0 \tag{1}$$ We can apply ...
2
votes
2answers
219 views

Is there any geometric explanation of relationship between Integral and derivative?

It is said integral is anti-derivative, derivative is tangent of graph function in each point on the function and integral is the area of the region in the xy-plane bounded by the graph. I can not ...
0
votes
1answer
42 views

Calculate the expected value

To get the expected value of $E(X), E(Y) $ and $E(X, Y)$ given: $$ f_{X,Y}(x,y) = 3x $$ where $0\le x \le y \le 1.$ My solution is, first get the margin distribution: \begin{aligned} f_x(x) &= ...
7
votes
0answers
197 views

How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing ...
0
votes
3answers
56 views

How do I solve $ \int_{-1}^{1} \cos^2(3\pi t) dt$

I have this integral: $$ \int_{-1}^{1}\cos^2(3 \pi t) dt$$ I don't remember the specific rules for solving this... I would find the primitive function to the above, but I don't know how to combine ...
0
votes
0answers
68 views

Integrating by exhibiting a partition

Is there some reasonable way of showing that $$\int_{a}^b\ c_0+c_1x+...+c_nx^n\ dx = c_0(b-a)+c_1\frac{(b^2-a^2)}{2}+...+c_n\frac{b^{n+1}-a^{n+1}}{n+1}, $$ by finding a partition of $[a,b]$ with lower ...
0
votes
2answers
33 views

Integrate a discrete function

how to know if you can integrate this function: $$ f(x) = \begin{cases} 1 & : \frac{1}{x},\ x \in \mathbb{Z}^+ \\ 0 & : \text{otherwise} \end{cases} $$ Over the ...
3
votes
1answer
36 views

Integral evaluation with exponentials

I want to evaluate the integral $\int_0^T e^{-ax}e^{-bx^2} \, dx$. I found a direct solution: $$\int_{0}^{\infty} e^{-ax}e^{-bx^2} \, dx = \sqrt\frac{\pi}{b} \exp\left(\frac{a^2}{4b}\right) ...
0
votes
1answer
23 views

A question about Integral on a bounded function

Is this statement correct: Let $F$ be a positive upper bounded function then for sufficiently small $\epsilon$ the function $F^\epsilon$ is bounded
1
vote
2answers
76 views

Evaluating $\;\int_{1}^{\ln3}\frac{e^x - e^{2x}}{(1 + e^x)}\,dx$

Find $\int_{1}^{\ln3}(e^x - e^{2x})/(1 + e^x)dx$. I looked through my notes for integration techniques and thought I could try a $u$ substitution but whatever I set $u$ to I can't seem to ...
0
votes
1answer
134 views

Integral $\int y\,e^{x^2}\,dy$

$$\int y\,e^{x^2}\,dy$$ I begin with $$\int e^{x^2}y\, dy$$ let $u=e^x$, $du=e^x\, dx$ how do I continue?
1
vote
3answers
51 views

Compute variance, using explicit PDF

I'm trying to get $\text{Var}(x)$ of $f(x) = 2(1+x)^{-3},\ x>0$. Please tell me if my working is correct and/or whether there is a better method I can use to get this more easily. $$ ...
2
votes
1answer
53 views

Find Pi using integral

I am just started learning calculus and wonder why: $$\int_0^1 \frac{4}{1+x^2}$$ Allows to find $\pi$? It would be great if someone could provide very detailed explanation.
0
votes
0answers
31 views

How to construct a function to map coefficients?

Surely this question is known by many people but I lack of enough maths knowledge so I prefer ask here. I have a triangular matrix that represent coefficients, all of them are rational numbers ...
0
votes
0answers
32 views

question about integration by substitution (for a general measure)

Let $\mu:=\mu(dx)$ be a probability measure on $\mathbb{R}$ such that $$\int_{\mathbb{R}}|x|\mu(dx)<+\infty$$ Define \begin{eqnarray} \rho: [0,+\infty]&\to&[0,+\infty) \\ ...
1
vote
3answers
60 views

Evaluate area of the field defined by $\left (\frac{x^2}{4}+y^2 \right )^2=x^2+y^2$

Evaluate the area of the field defined by $\left (\frac {x^2}{4}+y^2 \right )^2=x^2+y^2$ I tried to turn it into a function $y(x)$, bus I was unable to do it. Is it actually possible to solve it ...
1
vote
1answer
41 views

Integrate and derivative

i'm not able to explain the following step: $\frac{1}{k+v(x)}=\frac{d^2 v}{dx^2}$ by integrating this equation: $(C-\frac{1}{k+v(x)})^{\frac{1}{2}}=\frac{dv}{dx}$ Please, if somebody can help i'll ...
0
votes
2answers
40 views

Help finding k. Issue with integration

Let the continuous random variable $X$ have a probability density function $f(x)$ such that $$f(x) = k(1+x)^{-3}, x>0$$ $=0$ elsewhere Find k This is what I tried: $\int_0^\infty k(1+x)^{-3}dx ...
1
vote
2answers
30 views

question on derivation in book. integration

I do not see how the book went from the equation $(1-c^2) u_{\zeta\zeta}=\sin(u)$ to equation (5.2) below it. The books says to multiply both sides by $\frac{du}{d\zeta}$ and "integrate". I do not see ...
3
votes
0answers
45 views

Approximating this definite integral

I ran into the following integral in my research that I believe has no closed-form solution: $$ I = \int_{s_0}^{s_1} \frac{(\alpha_x s + \beta_x)^{\lambda_x}}{(\alpha_y s + \beta_y)^{\lambda_y}} ds ...
2
votes
1answer
65 views

$\int \frac{1}{y-1} dy = \ln |y - 1|$?

I read the following in a book on differential equations $$\int \frac{1}{y-1} dy = \log |y - 1|$$ If I put $\int \frac{1}{y-1} dy$ into Wolfram Alpha it gives $\log (y - 1)$, i.e. the argument of ...
1
vote
1answer
38 views

A certain relation of a polynomial to its coefficients

I've got a certain problem: If $A(t) = a_0+a_1t+ ...+a_Nt^N$, show that: $a_k = \frac{1}{2\pi}\int_{-\pi}^{\pi} e^{-ikx}A(e^{ix})dx$ after some rearrangements I got: $a_k = ...
2
votes
3answers
77 views

Indefinite integral of a rational function: $\int\frac{6x+4}{x^2+4}\,dx$

Find $\displaystyle\int\frac{6x+4}{x^2+4}\,dx$ The question asks to find integral of the expression so I divided them into two parts: $$ \int\frac{6x}{x^2+4}\,dx $$ and $$\int\frac{4}{x^2+4}. $$ So, ...
2
votes
3answers
216 views

Simplifying expression and finding indefinite integral

(a) Simplify $$\Large \frac{e^{-4x} + 3e^{-2x}}{e^{-4x}-9} \quad.$$ (b) Hence find $$\Large \int\frac{e^{-4x} + 3e^{-2x}}{e^{-4x}-9} \mathrm{d}x$$ I tried to find a breakdown of the expression, but ...
0
votes
1answer
23 views

Integral of Absolute of a Function

Suppose the function $f$ to be integrable on the closed interval $[a,b]$. Prove that the function $|f|$ is also integrable on that interval.
2
votes
0answers
43 views

Triple integral - volume of solid described by inequalities

I have to calculate the volume of solid described by inequalities: $$(x\leqslant y)\vee (y\leqslant z) \vee (x\leqslant z)$$ in region $[0,1]^3$. What is important, here we have conjunction. It is ...
0
votes
1answer
39 views

Bounding the average of a vector valued function

Disclaimer: I edited the question so that it fits Daniel Fischer's comments and it becomes more general. I also provide an answer myself in case anyone might be interested in the solution. Question: ...
0
votes
1answer
57 views

How to prove this? (Integration problem)

$$\sup_{x \in \mathbb{R}^{n}} \int_{\mathbb{R}^{n}} \frac{1}{|y|^{n-\alpha}}\frac{1}{1+|x-y|^{d+2}} dy < \infty,$$ For $0<d<\infty$ and $0< \alpha < n$.
2
votes
0answers
51 views

Multiple integral involving exponential and trigonometric functions

By using the generating function for Bessel functions I have discovered the following identity: \begin{eqnarray} &&\int\limits_{[0,2 \pi] \times [-\frac{\pi}{2},\frac{\pi}{2} ]^2} e^{\imath x ...
6
votes
4answers
288 views

Wolfram alpha says that $\int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$

Wolfram alpha says that $$ \int_{-\infty}^\infty e^{-ix^2}dx = \sqrt{\frac{\pi}{i}}$$ holds. But it has two different values ($\sqrt{i}$). How should I understand this?
1
vote
3answers
70 views

Calculating volume enclosed using triple integral

Calculate the volume enclosed between $x^2 + y^2=1$, $y^2+z^2=1$, $x^2+z^2=1$ I am supposed to solve this question using multiple integral. I am not able to visualize the resultant figure of ...
0
votes
0answers
76 views

Complex Gaussian Integral - $\int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}}$?

I found some formulas on books, especially the complex gaussian integral formula: $$ \int_{-\infty}^\infty e^{-p(t+c)^2}dt = \sqrt{\frac{\pi}{p}} $$ for $p,c\in\mathbb C$. Then if $p=i=\sqrt{-1}$, the ...
0
votes
1answer
40 views

help with logarithmic integration.

I've been googling some tutorials on integrating logarithms for my calc 2 class and I've found a lot of good stuff. Unfortunately nothing has answered how to handle a problem that I have. I've tried ...
4
votes
6answers
86 views

Given $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$

If $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$. Here is what I have done so far. I have took $f'(x)=(1+x^2)^{1/2}$ and I have found $1/f'(0)$ which should equal $1$. I don't think this ...
2
votes
2answers
60 views

Integral of $x^2 \cos(a x)\; \mathrm{d}x$

I am trying to solve the following problem: $\int x^2 \cos(a x)\; \mathrm{d}x$ I thought this would be simple and I am pretty sure this is the answer: $I ...
1
vote
1answer
78 views

Property of the Riemann Integral

Hello fellow Mathematics enthusiasts. I was hoping someone could help me with the following problem from Terry Tao's Introduction to Measure Theory: Let $[a,b]$ be an interval, and let $f,g:[a,b] \to ...
1
vote
1answer
55 views

How to integrate $\sqrt{1+(2/3)x}$?

How would you solve the following (step by step please!): $$\int^6_5\sqrt{1+\frac23x}\ dx$$ I started with $u=1+\frac23x$, $du=\frac23\,dx$, now what?