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)

1
vote
2answers
171 views

How to integrate logarithm and power function?

I am trying to solve the following integral $$\int_{0}^{1}(\ln(1+x))^2 x^{a-1}\,dx ; a>0.$$ I tried using partial functions but that didn't lead to anything. Any suggestion?
1
vote
1answer
54 views

I have a problem with solving this integral by partial fraction

I have an example like this : $\int \frac{1}{x^{4}}dx$ = $\int x^{-4}dx$ = $\frac{1}{-4 + 1}x^{-4 + 1}+C$ = $\frac{1}{-3}x^{-3}+C$ What if this? $\int \frac{1}{2x-10}dx$ = ?
1
vote
0answers
33 views

How to use the Riemann sum for a line integral

Let us say that we have the function $f(x,y) = xy$ and the curve $C: x=\cos(t), y=\sin(t)$ on the interval $[0, \pi/2]$. I want to use the Riemann sum to find out the integral of $$\int_{t=0}^{t=\frac{...
3
votes
1answer
64 views

How is this integral equal to this natural logarithm?

I am trying to understand following problem: $$\int {\sin x \over \cos x}dx = -\int {d \cos x \over \cos x} = - \ln \lvert \cos x\rvert + k $$ I don't really get the final step, are they equal ...
2
votes
3answers
103 views

Finding the area under the curve $x^{1/2} \sin x$ between $x=2$ to $x=10$

This question is with reference to my previously asked question. Which is the best way to find the area under the curve $x^{1/2}\sin{x}$ between $x=2$ to $x=10$ to the most accurate numerical value?
3
votes
2answers
299 views

Integration of x^(1/2) sinx

My book say that integration of $x^{1/2} \sin x$ is not possible, why is it so? Which functions do not have an anti derivative? Does it mean that they do not have any area under the curve? But that'...
0
votes
2answers
62 views

Evaluate the integral $\int_{-1}^1 \frac{1}{x^2-2x\cos\alpha+1}\mathrm dx,\alpha\in(0,\pi)$

For the equation $x^2-2x\cos\alpha+1=0$ solutions are $$x_1=\cos\alpha-\sqrt{\cos^2\alpha-1},x_2=\cos\alpha+\sqrt{\cos^2\alpha-1}\Rightarrow$$ $$\int_{-1}^1 \frac{1}{x^2-2x\cos\alpha+1}\mathrm dx=\...
1
vote
1answer
47 views

Change of variables in $\int_{\Omega}(x+y+z)^2xyz \: dxdydz$

I have the triple integral $\int_{\Omega}(x+y+z)^2xyz \: dxdydz$ , with $\Omega=\lbrace (x,y,z)\in \mathbb{R}^3:0\leq x+y+z\leq 1 , \:1\geq x\geq0, \: \: 1\geq y\geq 0, \: \:1\geq z\geq 0 \rbrace$. I ...
3
votes
3answers
72 views

How to evaluate the following integral, $\int\frac{x \, dx}{x^2+2x+17}$?

I am new to integration. This function is kinda tricky for me : $$\int\frac{x \, dx}{x^2+2x+17}$$ I came up with following three approaches: Partial fraction decomposition, but I can't factor the ...
3
votes
1answer
54 views

Convergence of an integral $\int_1^{+\infty} \frac{\ln^2(1+x)}{x^{2\alpha}}\mathrm dx$

$$\ln^2(1+x)\sim x^2-x^3,x\rightarrow \infty\Rightarrow \int_1^{+\infty} \frac{\ln^2(1+x)}{x^{2\alpha}}\mathrm dx=\int_1^{+\infty} \frac{x^2-x^3}{x^{2\alpha}}\mathrm dx=$$ $$\int_1^{+\infty} \frac{1}{...
3
votes
5answers
289 views

Book of integrals

Is there a book which contains just a bunch of integrals to evaluate? I want to learn new integration techniques and I'm open to other suggestions as to how I can go about learning new techniques. ...
0
votes
0answers
20 views

How to compute this integrale $\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2} } dx$

I would like to calculate the following integral $$I(a,y)=\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2}} dx, \quad a>0, y\in \mathbb R^3 .$$ Here's what I did: In ...
4
votes
2answers
72 views

Using 1-forms to integrate along curves

The following is written in my lecture notes: 'Suppose that $\gamma:[a,b]\rightarrow\mathcal{M}$ is a smooth curve and $\omega$ is a 1-form on $\mathcal{M}$. Then we get a smooth function $[t\...
1
vote
1answer
17 views

Problem in passage in proof from Willem's book: is that $h$ in $L^{p'}$? How else can I use Dominated Convergence if not?

Here is my problem: Lebesgue's dominated convergence theorem implies that $$\begin{align}\left|\int_{\Omega_+} n\,\eta'(n\,x_N)w\, u\, dx\right|&=\left|\int_{U\times]0,1/n[}n\,\eta'(n\,x_N)h\,...
1
vote
0answers
32 views

Integration on complex spheres and Gamma function

I'm studying special functions, especially Jacobi functions, related to the rank one groups ($U(1, n; \mathbb{F})$ where $\mathbb{F}$ is $\mathbb{C}$ or $\mathbb{H}$, the skew-field of quaternions), ...
3
votes
0answers
61 views

Prove the real-version of Riemann–Lebesgue lemma

I've been told to prove the real-version of Riemann–Lebesgue lemma, which is: for $f$ integrable and $2\pi$ periodic: $$ \lim_{n\to\infty} \int_{0}^{2\pi} f(x)\cos(nx) \ dx = \lim_{n\to\infty} \int_{...
0
votes
1answer
48 views

a Very simple example

When talking of functions that are Riemann integrable but not Lebesgue integrable we always give the example of $f(x)=\frac{\sin(x)}{x}$ on $]0,\infty [$ but my question is : is it the same with $f(x)...
0
votes
2answers
36 views

Evaluating $\iint \limits _R \cot^{-1} \frac y x {\rm d}A$ on $1 \le x^2 + y^2 \le 4, \ 0 \le y \le x$

I tried to evaluate $$\iint \limits _R \cot^{-1} \frac y x {\rm d}A,$$ where $R$ is bounded by $1 \le x^2 + y^2 \le 4$ and $0 \le y \le x$. How can I complete the solution?
3
votes
2answers
68 views

Another way to evaluate $\int_0^{\infty} x^2 e^{-x^2}dx$

In Stewart's Calculus book I came across the following Gaussian integral. Using $\int_{-\infty}^{\infty}\exp{(-x^2)}dx = \frac{\sqrt{\pi}}{2}$ evaluate $$ \int_0^{\infty} x^2 e^{-x^2}dx $$ I ...
1
vote
2answers
69 views

How was this integration problem $\int \frac{\sec^2 x}{\tan x}\:dx=\log |\tan x | +C$ solved?

$$\int \frac{\sec^2 x}{\tan x}\:dx=\log |\tan x | +C$$ When I started working on the problem, I used $1 + \tan^2x = \sec^2x$, but that didn't get me anywhere. How has the problem been solved?
3
votes
2answers
114 views

Calculating an integral $\int_{0}^{1}x^k(1-x)^{n-k}dx$

I confused for calculating $$\int_{0}^{1}x^k(1-x)^{n-k}dx$$ one solution that I guess is: $$1^{n}=(x+1-x)^{n}=\binom{n}{k}x^{k}(1-x)^{n-k}$$ so $$x^{k}(1-x)^{n-k}=\frac{1}{\binom{n}{k}}$$ finally $$\...
0
votes
0answers
20 views

Opposite of Monte Carlo

In this lecture, at 1:08:35, the lecturer goes from $$\text{argmin}\frac{1}{N}\sum\limits_{i=1}^{N}\text{log}\frac{p(x_i|\theta_0)}{p(x_i|\theta)}$$ to $$\text{argmin}\int\log\frac{p(x|\theta_0)}{...
0
votes
0answers
27 views

How to prove error of the composite trapezoidal rule which in series?

Composite trapezoidal rule with error in one term: Composite trapezoidal rule with error in series: The first formula is reasonable, but I can't understand the second one. I have looked through ...
0
votes
0answers
27 views

Switching limit and integral in a min function

Let $f_n$ and $f$ be probability density functions such that $f_n\rightarrow f$. Set $F_n(x)=\int_{-\infty}^x f_n(t)d\mu(t)$ and $F(x)=\int_{-\infty}^x f(t)d\mu(t)$. Define $l_n(x)=\min(f_n(x),f(x)...
4
votes
0answers
36 views

Does there exist such Borel measurable function? [duplicate]

Does there exist such Borel measurable function $f: \mathbb R \to [0,\infty )$ such that $\int _a ^b f \, dx = \infty$ for all $a< b $? My feeling is that such function should not exist, but I can'...
3
votes
1answer
43 views

Polar coordinates integration

Compute the following integrals over $R$ $f(x,y)\,dx\,dy$ over the area $R$ where: $f(x, y) = x$ and $R$ is given by $0 ≤ r ≤ \cos θ$ and $f(x, y) = x$. I understand polar coordinates is probably ...
1
vote
1answer
34 views

Distribution of a Poisson process with uniformly random parameter

Let $X = Unif(2, 4)$ and $Y=Poisson(X)$. My goal is to find $P(Y=n)$, but I always seem to get stuck on some nasty integral. Here's what I've tried: $P(Y=n) =\int_2^4P(Y=n|X=x)*P(X=x)dx = \int_2^4 ...
2
votes
0answers
70 views

Any good approximation for this integral?

I am interested in the following integral $$ \mathcal{I}=\int_{-\infty}^\infty\mathop{dz}\left[\frac{1}{\sqrt{a+b}(z^2)^{n/4}}-\frac{1}{\sqrt{a+b\cos^2\theta}(R^2+z^2)^{n/4}}\right], $$ where $R\ll 1$,...
-3
votes
2answers
91 views

Integration question (u-substitution) [closed]

$$ \int \limits _0^\pi \sin x\cos^2 x\ dx\,. $$
1
vote
1answer
27 views

Complex functions with local primitives

Problem from the book: "A Course in Complex Analysis" - Fisher|Lieb . Show that if a function $f:G\rightarrow \mathbb{C}$ has local primitive, then for every closed triangle $\Delta\subset G$ , we ...
6
votes
3answers
120 views

Evaluate the integral $\int x^{\frac{-4}{3}}(-x^{\frac{2}{3}}+1)^{\frac{1}{2}}\mathrm dx$

$$x^{\frac{-4}{3}}(-x^{\frac{2}{3}}+1)^{\frac{1}{2}}=\frac{\sqrt{(-\sqrt[3]{x^2}+1)}}{\sqrt[3]{x^4}}$$ Is it necessary to simplify the function further? What substitution is useful? $u=\sqrt[n]{\...
4
votes
1answer
40 views

Definite integrals that are hard using the FTC but doable from first principles

When I was introduced to integration, I was briefly exposed to the definition of a Riemann sum, and as an illustration we worked out a couple of definite integrals directly from this definition: $$ \...
2
votes
1answer
64 views

Evaluate the integral $\int \frac{1}{x^3\sqrt{x^2+x-1}}\mathrm dx$

What substitution to use in this integral? I tried to factorize $x^2+x-1$ and use a substitution $u=\sqrt{\frac{a(x-\alpha)}{x-\beta}}$.
1
vote
1answer
53 views

Integration by Substitution Rearranging

I have to show that the following the integral in terms of x can be rewritten as follows: $$\int_{-1}^1\frac{\sqrt{1-x^2}}{1+x^2}dx = \int_{u_2=}^{u_1=}\frac1{1+cos^2u}du-\pi$$ I have used the ...
4
votes
5answers
113 views

Solve the integral $\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx$

To find the Variance of a Wiener Process, $Var[W(t)]$, I have to compute the integral $$ Var[W(t)]=\dots=\frac 1 {\sqrt {2 \pi t}}\int_{-\infty}^{\infty} x^2 e^{-\frac {x^2} {2t}}dx=\dots=t. $$ I've ...
2
votes
5answers
188 views

Solve $ \int{\frac{7x^2 + 1}{(x+1)(x-1)(x+3)}}\,dx $ [closed]

I don’t know how to solve this integral: $$\int{\frac{7x^2 + 1}{(x+1)(x-1)(x+3)}}\,dx$$ I know this is a rational integral but I don’t know how to write it in a different way
2
votes
1answer
76 views

Compute the probability of a joint event involving two independent standard normals

Suppose $X$ and $Y$ are independent, standard normal random variables. I'm trying to compute the probability of the event $$ \{X \leq x, Y \leq kX\} $$ where $k$ is a positive constant. The ...
3
votes
2answers
95 views

Integrate expression with $x^6$ [duplicate]

So I'm trying to integrate this expression, but I'm not figuring out what's the best substitution to do... $ \int \frac {1}{x^6+1} dx $ I tried to take $x^6 +1 $ and write $ (x^2 + 1) (x^4 -x^2 + 1)...
1
vote
1answer
50 views

Small $\delta$ behavior of rational integral

I need to do the following integral, and extract the leading small $\delta$ behavior: $$I(\delta, A)=\int_0^1 dx \, x^n \frac{1}{(x^2 A+\delta)^\nu},\qquad 0< n < \nu, \enspace A\in\mathbb{R}$$...
-3
votes
1answer
46 views

A limit involving the gamma function [closed]

The limit is $$\displaystyle \lim_{n \to \infty} \frac1{2^{n/2}}\gamma\left(\frac{n}{2}\right) \int_{n+\sqrt{2n}}^\infty e^{-t/2} t^{n/2-1}\,dt$$ I am trying to compare it with the gamma function but ...
0
votes
0answers
90 views

evaluating Lebesgue integral 3

Let $A_{R,r}$ be a set with $0\lt r\lt R$ $$A_{R,r}= \{(x,y,z)\in\mathbb R^3\mid (R-\sqrt{x^2+y^2})^2+z^2\le r^2\}$$ Evaluate $\lambda_3(A_{R,r})$ Answer: I used Cavalieri's principle and set $$...
0
votes
1answer
67 views

Find the value of $\int_{7/2}^{11/2} \frac{f(x)}{x} dx$.

A decreasing, continuous function $f(x)$ satisfies the following three conditions: For all positive reals $x$, $f(x)>0$. For all positive reals $t$, the triangle with vertices $(0,0)$, $(...
2
votes
4answers
79 views

Using polar coordinates to find the area of an ellipse

Considering an ellipse with the $x$ radius equal to $a$ and the $y$ radius equal to b$:$ I figured that some kind of parameterization might be: $x=a\cos\theta$ $y=b\sin\theta$ and then polar $r^2$...
2
votes
2answers
101 views

Are all integrals an approximation?

Are all integrals an approximation of the result rather than $100$% accurate ? If so, why is $x^2$ the exact area under the curve of for each value of $f(x) = 2x$?
5
votes
3answers
118 views

Problem : $\int^{\pi/2}_0 \sin\theta \ln(\sin\theta)d\theta$ is equal to …

Problem : $\int^{\pi/2}_0 \sin\theta \ln(\sin\theta)d\theta$ is equal to (a) $\ln\frac{2}{e}$ (b) $-\ln\frac{2}{e}$ (c) $\ln 2$ (d) $1$ My approach : Let $I =\int^{\pi/2}_0 \sin\theta \...
6
votes
2answers
293 views

Counterexample of the almost-inverse of the Fundamnetal Theorem of Calculus(Lebesgue).

Can anyone give me a counterexample to the following statement: Suppose $F \colon [0,1] \to \mathbb{R}$ is continuous and differentiable almost everywhere, then $F(b)-F(a)=\int_a^b F'(t)\, \text{d}...
5
votes
2answers
60 views

Evaluate $\int \frac{x^2}{x^2 -6x + 10}\,dx$

Evaluate $$\int \frac{x^2}{x^2 -6x + 10} \, dx$$ I'd love to get a hint how to get rid of that nominator, or make it somehow simpler. Before posting this, I've looked into: Solve integral $\int{\...
1
vote
0answers
38 views

On the dog-bone contour around [-1,1], what are the arguments of these two lines approaching the real axis from above and below?

I am using a dog-bone contour to integrate around the interval [-1,1]. (-1 and +1 are branch points of the integrand.) I am using the principal branch of log, so I am restricting its argument to ...
0
votes
0answers
41 views

Integral by parts $\int_{0}^{2a}3\left(1+\frac{t^2}{4}\right)\frac{1-\cos t}{2}\frac{1}{2}\,dt$

How do I solve this integral by parts? $$\int_{0}^{2a}3\left(1+\frac{t^2}{4}\right)\frac{1-\cos t}{2}\frac{1}{2}\,dt$$ thank you
5
votes
0answers
77 views

How do I evaluate this integral :$ \int_0^\frac{\pi}{2} [\text{chi}(\cot^2x)+\text{shi}(\cot^2x)]\csc^2(x)e^{-\csc^2(x)}dx$?

I have tried to evaluate this integral :$$ \int_0^\frac{\pi}{2} [\text{chi}(\cot^2x)+\text{shi}(\cot^2 x)]\csc^2(x)e^{-\csc^2(x)}dx $$ where, $\text{shi}(x)=\int_{0}^{x}\frac{\sinh t}{t}dt$ , $\text{...