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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4
votes
2answers
3k views

Integrate $\csc^3{x} \ dx$

I found these step which explain how to integrate $\csc^3{x} \ dx$. I understand everything, except the step I highlighted below. How did we go from: $$\int\frac{\csc^2 x - \csc x \cot x}{\csc x - \...
3
votes
6answers
360 views

Limit $\lim_{n\to \infty} \frac{1}{2n} \log{2n\choose n}$ [duplicate]

$\lim_{n\to \infty} \frac{1}{2n} \log{2n\choose n}$ I could not approach it beyond these simple steps, $\lim_{n\to \infty} \frac{1}{2n} \log(\frac{2n!}{(n!)^2})$ $=\lim_{n\to \infty} \frac{1}{2n}...
2
votes
2answers
524 views

Calculating: $\lim_{n\to \infty}\int_0^\sqrt{n} {(1-\frac{x^2}{n})^n}dx$ [duplicate]

Possible Duplicate: Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$ I need some help calculating the above limit. What i have ...
7
votes
3answers
120 views

Integral of $\frac{\sqrt{x^2+1}}{x}$

So I have to do the integration of $\frac{\sqrt{x^2+1}}{x}$. Give me a hint. What should I replace? Should I do it with integration by parts?
5
votes
0answers
360 views

Integral of $\frac{1}{x^4+1}$ [duplicate]

Just doing this for revision, seems much harder than it should be, should I use $x=\tan u$ ? Any help appreciated.
3
votes
1answer
89 views

Evaluating $\int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx$

Let $\alpha \in \mathbb R$ and let's call $I:=[0,1]$. Evaluate $$ \int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx. $$ Well, the case $n=1$ is easy and the integral equals $\frac{...
3
votes
2answers
401 views

How can I show that $f$ must be zero if $\int fg$ is always zero?

Let $f(x)$ be continuous on $[a,b]$ and suppose $\int_a^b f(x)g(x)dx = 0$ for every continuous function $g$ on $[a,b]$. Prove that $f(x)=0$ on $[a,b]$. I understand that $f(x)$ must be zero ...
3
votes
2answers
710 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 ...
2
votes
1answer
108 views

$C[0,1]$ with $L^1$ norm is not Banach space.

I want to check that $(C[0,1],∥⋅∥_1)$ is not a Banach space, where $\|f\|_1 = \int_0^1 |f(x)|\,{\rm d}x$.I took $(f_n)_{n \geq 1}$ a sequence in $C[0,1]$ given by:$f_n: [0,1] \rightarrow \mathbb{R}, \...
2
votes
1answer
260 views

About the confluent versions of Appell Hypergeometric Function and Lauricella Functions

I know there are two important properties about Appell Hypergeometric Function and Lauricella Functions: $F_1(a,b_1,b_2,c;x,y)=\dfrac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-xt)...
0
votes
6answers
3k views

Evaluating $\int \sqrt{1 + t^2} dt$?

How do we solve (i.e. get the closed form of) $\int \sqrt{1 + t^2} dt$ ? The Wolfram page shows the closed form of it but not the steps in solving it. I think I need some algebraic trick.. I ...
46
votes
4answers
4k views

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx.$$ I do not have a strong reason to be sure it exists, but I ...
41
votes
2answers
3k views

Integration of forms and integration on a measure space

In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus): the ...
55
votes
2answers
2k views

Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$

Please help me to find a closed form for the following integral: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$ I was told it could be calculated in a ...
38
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 2\...
33
votes
4answers
3k views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
31
votes
1answer
1k views

Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$

How can I prove the following identity? $$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$$
35
votes
7answers
3k views

Evaluating the integral, $\int_{0}^{\infty} \ln\left(1 - e^{-x}\right) \,\mathrm dx $

I recently got stuck on evaluating the following integral. I do not know an effective substitution to use. Could you please help me evaluate: $$\int_{0}^{\infty} \ln\left(1 - e^{-x}\right) \,\mathrm ...
11
votes
1answer
746 views

What would qualify as a valid reason to believe there is a closed form?

I noticed that almost every non-homework-level integral posted on this site prompts somebody to ask "Do you have any reason to believe there is a closed form?" (some recent examples here and here) I ...
23
votes
4answers
787 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it known any ...
27
votes
4answers
1k 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
988 views

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = \...
32
votes
5answers
1k views

Evaluating $\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$

Some time ago I came across to the following integral: $$I=\int_{0}^{1}\frac{1-x}{1+x}\frac{\mathrm dx}{\ln x}$$ What are the hints on how to compute this integral?
20
votes
1answer
621 views

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

I am trying to prove that $$\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx = -\frac{\pi^3}{48}-\frac{\pi}{8}\log^2 2 +G\log 2$$ where $G$ is the Catalan's Constant. Numerically, it's ...
19
votes
2answers
704 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 ...
22
votes
4answers
2k views

Good book on evaluating difficult definite integrals (without elementary antiderivatives)?

I am very interested in evaluating difficult definite integrals without elementary antiderivatives by manipulating the integral somehow (e.g. contour integration, interchanging order of integration/...
25
votes
1answer
508 views

Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$

Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: $$\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)...
53
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 ...
19
votes
2answers
509 views

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
13
votes
1answer
1k views

A Putnam Integral $\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$

This is a Putnam Problem that I have been trying to solve (on and off) for two years, but I have failed. I am in Calculus BC. This problem comes from the book "Calculus Eighth Edition by Larson, ...
13
votes
4answers
499 views

Finding $\int_0^{\frac{\pi}{2}}\arctan\left(\sin x\right)dx$

$$\int_0^{\frac{\pi}{2}}\arctan\left(\sin x\right)dx$$ I try to solve it, but failed. Who can help me to find it? I encountered this integral when trying to solve $\displaystyle{\int_0^\pi\frac{x\...
10
votes
6answers
826 views

show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$

show that $$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$ using different ways thanks for all
26
votes
4answers
3k views

When is an elliptic integral expressible in terms of elementary functions?

After seeing this recent question asking how to calculate the following integral $$ \int \frac{1 + x^2}{(1 - x^2) \sqrt{1 + x^4}} \, dx $$ and some of the comments that suggested that it was an ...
26
votes
4answers
6k views

Using Integration By Parts results in 0 = 1

I've run into a strange situation while trying to apply Integration By Parts, and I can't seem to come up with an explanation. I start with the following equation: $$\int \frac{1}{f} \frac{df}{dx} ...
8
votes
3answers
729 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 ...
16
votes
5answers
51k 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 ...
14
votes
3answers
536 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 $\dfrac{\sin^{-1}x}{\sqrt{1-x^...
11
votes
4answers
715 views

Volume of $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$

Let $T_n=\{x_i\ge0:x_1+\cdots+x_n\le1\}$. I know $T_n$ is tetrahedron. My question: How can I compute the volume of $T_n$ for every $n$?
10
votes
1answer
169 views

To find the minimum of $\int_0^1 (f''(x))^2dx$

I was trying to solve a question of an entrance exam. I am completely stuck in the problem. I am not able to find idea how to proceed. Please help me. Let $A$ be the set of twice continuously ...
9
votes
2answers
4k views

List of functions not integrable in elementary terms

When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the ...
18
votes
3answers
630 views

Find functions family satisfying $ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$

I wonder what kind of functions satisfy $$ \lim_{n\to\infty} n \int_0^1 x^n f(x) = f(1)$$ I suppose all functions must be continuous.
12
votes
5answers
29k 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 ...
11
votes
1answer
442 views

Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx $

Prove the following $$\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx = -3\zeta(5)+\pi^2 \frac{\zeta(3)}{3}$$ where $$\operatorname{Li}^2_2(x) =\left(\int^x_0 \frac{\log(1-t)}{t}\,dt \right)^2$$
9
votes
2answers
508 views

How to evaluate this integral?

How would I prove that$$\int_{0}^\infty \frac{\cos (3x)}{x^2+4}dx= \frac{\pi}{4e^6}$$ I changed it to $$\int_{0}^\infty \frac{\cos (3z)}{(z+2i)(z-2i)}dz$$, and so the two singularities are $2i$ and $-...
13
votes
5answers
2k views

Question about Riemann integral and total variation

Let $g$ be Riemann integrable on $[a,b]$, $f(x)=\int_a^x g(t)dt $ for $x \in[a,b]$. Can I show that the total variation of $f$ is equal to $\int_a^b |g(x)| dx $?
12
votes
3answers
1k views

The equivalence between Cauchy integral and Riemann integral for bounded functions

Definitions Suppose $P\colon a=x_0<x_1<\dotsb<x_n=b$ is a partition of $[a,b]$. Let $\Delta x_k=x_k-x_{k-1}$ and $\lVert P\rVert$ denotes $\max_{0<k\le n}\Delta x_k$. The Cauchy integral ...
9
votes
2answers
544 views

If $\int_0^\infty {e^{-\lambda t}f(t){\rm d}t} = 0$ for all $\lambda >0$ then $f=0$ a.e.?

I am studying a paper in which the author uses something like that: Let $f$ be a bounded and Lebesgue measurable function. If $$\int_0^\infty {e^{-\lambda t}f(t)\,{\rm d}t} = 0 \qquad\text{for all} ...
7
votes
1answer
15k views

Understanding the relationship between differentiation and integration

I am trying to understand the relationship between differentiation and integration. Differentiation has been introduced to me by this diagram: Which displays that the derivative of a point $x$ on ...
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} \text{d}t=\int_{0}^{\frac{\pi}{2}...
10
votes
2answers
652 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 ...