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 ...

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6
votes
3answers
408 views

Improper integration involving complex analytic arguments

I am trying to evaluate the following: $\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$ Any help will be much appreciated.
15
votes
4answers
38k views

Is there a rule of integration that corresponds to the quotient rule?

When teaching the integration method of u-substitution, I like to emphasize its connection with the chain rule of integration. Likewise, the intimate connection between the product rule of derivatives ...
13
votes
6answers
584 views

How closely can we estimate $\sum_{i=0}^n \sqrt{i}$

By looking at an integral and bounding the error?
10
votes
6answers
2k views

How do I show that $\int_{-\infty}^\infty \frac{ \sin x \sin nx}{x^2} \ dx = \pi$?

Quick question. Could somebody please explain to me why it is that $$\int_{-\infty}^\infty \frac{ \sin x \sin nx}{x^2} \ dx = \pi$$ for every positive integer $n$? This integral showed up when I was ...
9
votes
2answers
514 views

Gaussian Integral

Consider the following Gaussian Integral $$I = \int_{-\infty}^{\infty} e^{-x^2} \ dx$$ The usual trick to calculate this is to consider $$I^2 = \left(\int_{-\infty}^{\infty} e^{-x^2} \ dx \right) ...
8
votes
3answers
8k views

How do you integrate a Bessel function? I don't want to memorize answers or use a computer, is this possible?

I am attempting to integrate a Bessel function of the first kind multiplied by a linear term: $\int xJ_n(x)\mathrm dx$ The textbooks I have open in front of me are not useful (Boas, Arfken, various ...
6
votes
2answers
619 views

Is there any notable difference between studying the Riemann integral over open intervals and studying it over closed intervals?

(1) A function $f:[a,b]\to\mathbb{R}$ is said to be Riemann integrable on $[a,b]$ . . . (2) A function $f:(a,b)\to\mathbb{R}$ is said to be Riemann integrable on ...
10
votes
1answer
215 views

How do I calculate the triple integral $\int_{0}^{2\pi}\int_0^1 \int_0^1 xy\sqrt{x^2 + y^2 -2xy\cos(\theta)} dx \text{ }dy \text{ } d\theta$?

I have tried a couple substitutions which didn't pan out, and I tried differentiating under the integral sign on $$ I(\theta) = \int_0^1 \int_0^1 xy\sqrt{x^2 + y^2 -2xy\cos(\theta)} dx \text{ }dy $$ ...
10
votes
2answers
502 views

Families of functions closed under integration

What are some concrete families $\mathcal F$ of real functions that are closed under integration in the sense that for every $f \in \mathcal F$ there is $F \in \mathcal F$ such that $F'=f$? Here are ...
10
votes
2answers
12k views

Problem when integrating $e^x / x$.

I made up some integrals to do for fun, and I had a real problem with this one. I've since found out that there's no solution in terms of elementary functions, but when I attempt to integrate it, I ...
8
votes
4answers
493 views

Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$

Hi I am trying to prove$$ I:=\int_0^1 \log\left(\,\Gamma\left(x+\alpha\right)\,\right)\,{\rm d}x =\frac{\log\left(2\pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha\,,\qquad \alpha \geq 0. $$ I am ...
8
votes
3answers
208 views

Convergence/Divergence of $\int_{0}^{1/e} \frac{\log \left(\frac{1}{x}\right)}{(\log^2 (x)-1)^{3/2}} \mathrm{dx}$

Initially I wanted to compute $$\int_{0}^{1/e} \frac{\log \left(\frac{1}{x}\right)}{(\log^2 (x)-1)^{3/2}} \mathrm{dx}$$ but it seems that Mathematica says that the integral diverges. I thought of ...
3
votes
1answer
2k views

Proving Abel-Dirichlet's test for convergence of improper integrals using Integration by parts

I'm struggling with the following calculus question. Let there be two functions $f,g : [a, \infty) \to \mathbb R$ such that: $g$ is monotonic, differentiable and has a limit at zero $f$ is ...
2
votes
5answers
16k views

how to calculate the integral of $\sin^2(x)/x^2$ [duplicate]

Possible Duplicate: Proof for an integral involving sinc function How do I show that $\int_{-\infty}^\infty \frac{ \sin x \sin nx}{x^2} \ dx = \pi$? ...
11
votes
4answers
2k views

Integral:$ \int^\infty_{-\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}dx $

How to evaluate: $$ \int^\infty_{-\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}dx $$ Maybe we can evaluate it using the well-known result:$\int_{0}^{\frac{\pi}{2}} \ln{\sin t} ...
7
votes
1answer
3k views

The absolute value of a Riemann integrable function is Riemann integrable.

This is an exercise in Bartle & Sherbert's Introduction to Real Analysis second edition. They ask to show that if $I=[a,b]$ is a closed bounded interval and that $f:I\to\mathbb{R}$ is (Riemann) ...
7
votes
5answers
516 views

A little integration paradox

The following integral can be obtained using the online Wolfram integrator $$ \int \frac{dx}{1+\cos^2 x} = \frac{\tan^{-1}(\frac{\tan x}{\sqrt{2}})}{\sqrt{2}}$$ Now assume we are performing this ...
6
votes
0answers
316 views

Finding $\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$ if it exists [duplicate]

Does there exist the following limitation? If the answer is yes, could you show me how to find that? $$\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$$ In the following, I'm going to write what ...
6
votes
1answer
198 views

Understanding the solution of $\int\left(1-x^{p}\right)^{\frac{n-1}{p}}\log\left(1-x^{p}\right)dx$

I would like to solve the integral $$\int\left(1-x^{p}\right)^{\frac{n-1}{p}}\log\left(1-x^{p}\right)dx.$$ My problem is that I arrived at a solution via wolfram alpha, but I would like to ...
3
votes
1answer
240 views

How can one prove the impossibility of writing $ \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions?

Can we express $ \displaystyle \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions? (Note: Infinite series are not allowed.) If not, then is there a proof that $ \displaystyle \int ...
0
votes
3answers
149 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
7
votes
2answers
1k views

Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus

This refers back to the integral of $\frac{\sin(x)}x = \frac\pi2$ already posted. How do I arrive at $\frac\pi2$ using the residue theorem? I'm at the following point: $$\int \frac{e^{iz}}{z} - \int ...
4
votes
1answer
158 views

Integral calculus proof

If $f(x)$ is continuous in $[a,b]$, prove that $ \displaystyle \lim_{n \to \infty} \dfrac{b-a}{n} \displaystyle \sum^n _{k=1} f\left( a + \dfrac{k(b-a)}{n} \right) = \displaystyle \int_a ^ b f(x)dx$ ...
4
votes
3answers
848 views

Integrating $\frac{\log(1+x)}{1+x^2}$ [duplicate]

Possible Duplicate: Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx$ I am a bit stuck here in evaluating the following integral:$$\int_{0}^{1}\frac{\log(1+x)}{1+x^2}\,\mathrm ...
3
votes
2answers
240 views

Integrating by substitution

I'm embarrassed to ask this question, but what's the flaw in the following evaluation? $\displaystyle\int_{0}^{\pi} \sin (\sin x) \ dx = \int_{0}^{0} \frac{\sin u}{\sqrt{1-u^{2}}} \ du = 0$.
2
votes
1answer
245 views

Is this definition missing some assumptions?

In his "Calculus on manifolds" Spivak first defines $n$-dimensional (Riemann-) integral over rectancles, then over Jordan measurable subsets of rectangles and finally extends it to open sets using ...
1
vote
3answers
167 views

how to solve $\int\frac{1}{1+x^4}dx$ [duplicate]

i want find the answer and metod of solve of $\int\frac{1}{1+x^4}dx$. I know $$\int\frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan\frac{x}{a}+C$$, How I can use this to solve of that integration.
1
vote
1answer
150 views

Integral of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$F(z)=\sum_{n=1}^\infty a_nz^n$$ converges in $|z|<1$. How can I ...
1
vote
2answers
781 views

How do you integrate $e^{x^2}$?

I know that $\int{\frac{1}{x}}dx$ is simply $\ln{(x)}+c$ (-which is clearly unrelated to the problem but I just thought I would share anyway) but I am not sure how to approach $e^{x{^2}}$. Perhaps a ...
11
votes
5answers
644 views

Why is $\int^\infty_{-\infty} \frac{x}{x^2+1} dx$ not zero?

We break up $\int^\infty_{-\infty} \dfrac{x}{x^2+1} dx$ into: $$\lim_{t\to -\infty} \int^0_t \dfrac{x}{x^2+1} dx + \lim_{t\to \infty} \int^t_0 \dfrac{x}{x^2+1} dx$$ So, evaluated, this gives; ...
10
votes
5answers
2k views

Is $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dy$?

Is it true that $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dy$ ? Just making sure. If not, how about: $\int_a^b f(x) dx = (f(b)-f(a))b - \int_{f(a)}^{f(b)}f^{-1}(x)dx$ ? I'm having a hard ...
5
votes
2answers
340 views

Trig integral $\int{ \cos{x} + \sin{x}\cos{x} dx }$

Assume we have: $ \int{ \cos{x} + \sin{x}\cos{x} dx } $ 2 ways to do it: Use $\sin{x}\cos{x} = \frac{ \sin{2x} }{2} $ Then $ \int{ \cos{x} + \frac{\sin{2x}}{2} dx } $ $ = \sin{x} - \frac{ cos{2x} ...
2
votes
4answers
740 views

Using Spherical coordinates find the volume:

Inside the surfaces $z=x^2+y^2$ and $z=\sqrt{2-x^2-y^2}$ I integrated over the ranges: $0 \leq \theta \leq 2\pi$ $ 0 \leq \phi \leq \frac{\pi}{2}$ $0 \leq r \leq \sqrt{2}$ I get ...
2
votes
3answers
119 views

Fractional Trigonometric Integrands

$$∫\frac{a\sin x+b\cos x+c}{d\sin x+e\cos x+f}dx$$ $$∫\frac{a\sin x+b\cos x}{c\sin x+d\cos x}dx$$ $$∫\frac{dx}{a\sin x+\cos x}$$ What are the relations between the numerator in the denominator, and ...
2
votes
4answers
23k views

How to integrate $\frac{1}{\sqrt{1+x^2}}$ using substitution?

How you integrate $\frac{1}{\sqrt{1+x^2}}$ using following substitution? $1+x^2=t$ $\Rightarrow$ $x=\sqrt{t-1} \Rightarrow dx = \frac{dt}{2\sqrt{t-1}}dt$... Now I'm stuck. I don't know how to proceed ...
2
votes
2answers
380 views

Insidious exponential integral

I hope that someone's up for the challenge; I'm attempting to solve this via computer: \begin{equation} \int_{-\pi}^\pi{\displaystyle \frac{e^{i\cdot a\cdot t}(e^{i\cdot b\cdot t}-1)(e^{i\cdot c ...
1
vote
0answers
80 views

Validity of Substitutions in Integrals [duplicate]

In integrals, we often make 'u' substitutions, which involve writing $u=f(x)$, and then also writing $du=f'(x)dx$. How valid is this? I've heard that you aren't allowed to do things like 'break' up ...
1
vote
1answer
147 views

Integral remainder converges to 0

I want to show that $\displaystyle \log(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$ for $-1<x\leq1$ i want to show it with the integral remainder of the taylor series that gave me: ...
1
vote
1answer
474 views

Limit of $ (\int_a^b f(x)^n \ dx)^{1/n}$ when $n\to\infty$

I've been working on the following problem: Show that if $f\in C[a, b]$ , $f\ge 0$ on $[a, b]$, then $\left(\int_a^b f(x)^n \,dx\right)^{1/n}$ converges when $n\to\infty$ and the limit is ...
0
votes
1answer
55 views

Generalized Riemann Integral: Improper Version

Reference For a bounded nonexample of integrability see: Riemann Integral: Bounded Nonexample For a convergence theorem on integral see: Riemann Integral: Uniform Convergence For a comparison of ...
0
votes
1answer
174 views

Inverse Fourier transform to find out $\hat c_1$

If we have an integration which is need to solve inversely $$a_0 e^{-r^2/R^2} = \int_0^\infty \hat{c}_1(k) \frac{\sin(k r)}{r} dk,$$ If I transform the $\sin(kr)$, then we get imaginary part. Please ...
0
votes
1answer
210 views

On the convergence of a specific sequence of integrable functions

Let $\{f_n\}$ a sequence of measurable non-negative functions on $\mathbb{R}$ converging point-wise on $\mathbb{R}$ to $f$, and let $f$ integrable over $\mathbb{R}$. If $\displaystyle ...
43
votes
2answers
1k views

Ramanujan log-trigonometric integrals

I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} ...
33
votes
2answers
2k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos ...
35
votes
2answers
994 views

Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$

Here is another integral I'm trying to evaluate: $$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx.\tag1$$ A numeric approximation is: ...
33
votes
1answer
618 views

Generalizing $\int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{\operatorname dx}{x^{2}+1} = \frac{5\pi^{2}}{96}$

The following integral \begin{align*} \int_{0}^{1} \frac{\arctan\sqrt{x^{2} + 2}}{\sqrt{x^{2} + 2}} \, \frac{dx}{x^{2}+1} = \frac{5\pi^{2}}{96} \tag{1} \end{align*} is called the Ahmed's integral ...
29
votes
3answers
848 views

Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $

How to prove $$ \int_{0}^{\pi/2}\ln\left(\,x^{2} + \ln^{2}\left(\,\cos\left(\,x\,\right)\,\right) \,\right)\,{\rm d}x\ =\ \pi\ln\left(\,\ln\left(\, 2\,\right)\,\right) $$ I don't know how to ...
22
votes
4answers
929 views

Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$

Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$ ...
29
votes
3answers
805 views

Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$

Here is a challenging one maybe some would like a go at. Show that: ...
54
votes
14answers
9k views

Why do we need to learn integration techniques?

After a lifetime of approaching math the wrong way, I took two college math courses this quarter with a newfound zest for math. These classes are integral calc and multivariable calc. Integral calc ...