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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5
votes
2answers
332 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
647 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
2answers
378 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 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
1answer
140 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: ...
0
votes
1answer
53 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
169 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
206 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 ...
38
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} ...
70
votes
2answers
2k views

Some users are mind bogglingly skilled at integration. How did they get there?

Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this. Especially in a ...
36
votes
8answers
1k views

Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$

Evaluate $$\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$$
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 ...
27
votes
2answers
909 views

Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?

This post of Boris Bukh mentions amazing Gustav Herglotz's integral $$\int_0^1\frac{\ln\left(1+t^{\,4\,+\,\sqrt{\vphantom{\large A}\,15\,}\,}\right)}{1+t}\ \mathrm ...
35
votes
2answers
944 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
586 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 ...
22
votes
4answers
895 views

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

Show that $\displaystyle{\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2} = \frac{\pi}{2}\log \left(\sqrt{2\pi} \Gamma\left(\frac{3}{4}\right) / \Gamma\left(\frac{1}{4}\right)\right)}$ ...
27
votes
3answers
774 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 ...
27
votes
3answers
760 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: ...
18
votes
3answers
1k views

How do I evaluate this integral $\int_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x$?

Show that $$\int\limits_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x =\frac{{{\pi^3}}}{{15}}+2\pi \ln^2 \left({\frac{{1+\sqrt 5 }}{2}}\right).$$ I don't have any idea how to start, ...
52
votes
14answers
8k 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 ...
33
votes
4answers
1k views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
27
votes
2answers
857 views

Show $\int_{0}^{\frac{\pi}{2}}\frac{x^{2}}{x^{2}+\ln^{2}(2\cos(x))}dx=\frac{\pi}{8}\left(1-\gamma+\ln(2\pi)\right)$

Here is an interesting, albeit tough, integral I ran across. It has an interesting solution which leads me to think it is doable. But, what would be a good strategy?. ...
22
votes
8answers
1k views

A proof of $\int_{0}^{1}\left( \frac{\ln t}{1-t}\right)^2\,\mathrm{d}t=\frac{\pi^2}{3}$

What is the proof of the following: $$\int_{0}^{1} \left(\frac{\ln t}{1-t}\right)^2 \,\mathrm{d}t=\frac{\pi^2}{3} \>?$$
19
votes
4answers
933 views

Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx = \frac{\pi^3}{16}-3G\log 2 \tag{1}$$ where $G$ is Catalan's Constant. I was able to express it in terms of ...
19
votes
4answers
923 views

Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$

Background: Evaluation of $\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$ We can prove using the Beta-Function identity that $$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma ...
23
votes
4answers
987 views

Integral $\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}$

Hi I am trying to show$$ I:=\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}. $$ Thank you. What a desirable thing to want to prove! It is a ...
45
votes
3answers
1k views

How much does symbolic integration mean to mathematics?

(Before reading, I apologize for my poor English ability.) I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast ...
18
votes
8answers
6k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
15
votes
2answers
545 views

Integral $\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.

I am trying to calculate $$ I=\frac{1}{\pi}\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta=\frac{11\pi^4}{180}=\frac{11\zeta(4)}{2}. $$ Note, we can expand the log in the integral to ...
8
votes
2answers
323 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
18
votes
7answers
995 views

A simple way to evaluate $\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx$?

I am currently trying to show that $\int_{-\infty}^\infty \cos(x^2) \, \mathrm dx = \sqrt{\frac{\pi}{2}}$ and the last integral I have to evaluate is $$\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx.$$ ...
16
votes
8answers
1k views

$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$ Evaluate Integral

Compute the following integral: $$\int_{0}^{\infty} \frac{e^{-x} \sin(x)}{x} dx$$ Any hint, suggestion is welcome. Thanks.
15
votes
4answers
910 views

How to evaluate $\int_{0}^{2\pi}e^{\cos \theta}\cos( \sin \theta) d\theta$?

For $\alpha \in \mathbb{R}$, define $\displaystyle I(\alpha):=\int_{0}^{2\pi}e^{\alpha \cos \theta}\cos(\alpha \sin \theta)\; d\theta$. Calculate $I(0)$. Hence evaluate ...
12
votes
3answers
456 views

integral of $\int \limits_{0}^{\infty}\frac {\sin (x^n)} {x^n}dx$

what is the answer of $$\int \limits_{0}^{\infty}\frac {\sin (x^n)} {x^n}dx$$ I saw the answer of $$\int \limits_{0}^{\infty}\left(\frac {\sin x} {x}\right)^ndx$$ but for my question i didn't see ...
12
votes
6answers
872 views

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

Can this integral be solved with contour integral or by some application of residue theorem? $$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$ It has two ...
15
votes
2answers
543 views

Evaluating $\int_{-1}^{1}\frac{\arctan{x}}{1+x}\ln{\left(\frac{1+x^2}{2}\right)}dx$

This is a nice problem. I am trying to use nice methods to solve this integral, But I failed. $$\int_{-1}^{1}\dfrac{\arctan{x}}{1+x}\ln{\left(\dfrac{1+x^2}{2}\right)}dx, $$ where ...
14
votes
5answers
629 views

Evaluate the integral $\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$

The problem is : Evaluate the integral $$\int_{0}^{\infty} \frac{1}{(1+x^2)\cosh{(ax)}}dx$$ I have tried expand $\frac{1}{\cosh{ax}}$ and give the result in the following way: First, note ...
9
votes
4answers
448 views

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting ...
12
votes
4answers
652 views

Computing $ \int_a^b \frac{x^p}{\sqrt{(x-a)(b-x)}} \mathrm dx$

I would like to compute the integral: $$ \int_a^b \frac{x^p}{\sqrt{(x-a)(b-x)}} \mathrm dx$$ where $$ a<b$$ and $$ p\in\mathbb{N}$$ $$ \int_a^b \frac{x^p}{\sqrt{(x-a)(b-x)}} \mathrm ...
11
votes
4answers
441 views

How to find $\int_{0}^{1}\dfrac{\ln^2{x}\ln^2{(1-x)}}{2-x}dx$

How to find $$ I=\int_{0}^{1}{\ln^{2}\left(x\right)\ln^{2}\left(1 - x\right) \over 2 - x} \,{\rm d}x $$ My idea: Let $x=1-t$, then $$ I =\int_{0}^{1}{\ln^{2}\left(1 - ...
10
votes
3answers
20k views

How to calculate the integral in normal distribution?

The factory is making products with this normal distribution: $\mathcal{N}(0, 25)$. What should be the maximum error accepted with the probability of 0.90? [Result is 8.225 millimetre] How will I ...
20
votes
5answers
790 views

Computing $\lim\limits_{n\to+\infty}n\int_{0}^{\pi/2}xf(x)\cos ^n xdx$

I got stuck at the following problem. Let $f\in C([0,\pi/2])$, then compute $$ \lim_{n\to+\infty}n\int\limits_{0}^{\pi/2}xf(x)\cos ^n xdx $$ Could you suggest a helpful idea?
14
votes
2answers
537 views

A closed form of $\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$$ I've tried substitution $y={1}/{x}$ and $e^y={1}/{x}$, but those didn't help ...
12
votes
4answers
467 views

Convergence $I=\int_0^\infty \frac{\sin x}{x^s}dx$

Hi I am trying to find out for what values of the real parameter does the integral $$ I=\int_0^\infty \frac{\sin x}{x^s}dx $$ (a) convergent and (b) absolutely convergent. I know that the integral ...
8
votes
3answers
665 views

Compute $\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space$

How would you approach $$\int_0^{\pi/2}\frac{\sin 2013x }{\sin x} \ dx\space?$$ The way I see here involves Dirichlet kernel. I wonder what else can we do, maybe some easy/elementary approaching ...
7
votes
4answers
449 views

How to compute $\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$

$$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$ WolframAlpha gives a numerical answer of $43.8122$, which appears to be $\sqrt{611\pi}$. And playing with that, it seems ...
4
votes
4answers
277 views

Evaluating $\int^1_0 \frac{\operatorname{Li}_3(x)}{1-x} \log(x)\, \mathrm dx$

How would you solve the following $$\int^1_0 \frac{\operatorname{Li}_3(x)}{1-x} \log(x)\, \mathrm dx$$ I might be able to relate the integral to Euler sums .
12
votes
5answers
20k views

Why does $1/x$ diverge?

So for the formula $\dfrac {1}{x}$, If you were to add up all $y$ values from $x=1$ to $x=∞$, wouldn't the sum approach a number because even though you are always adding, aren't you just adding ...
10
votes
3answers
397 views

Evaluating a sum involving binomial coefficient in denominator

I came across the following sum: $$\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}\frac{4^k}{{2k \choose k}}$$ I thought that this can be evaluated using the expansion of ...