Questions on the evaluation of definite and indefinite integrals
104
votes
3answers
5k views
The Integral that Stumped Feynman?
In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods ...
66
votes
5answers
3k views
Help find hard integrals that evaluate to $59$?
My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$.
I want to try and make a definite integral that equals $59$. So ...
53
votes
3answers
2k views
Is there an integral that proves $\pi > 333/106$?
The following integral,
$$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$
is clearly positive, which proves that $\pi < 22/7$.
Is there a similar integral which proves ...
50
votes
2answers
732 views
Two curious “identities” on $x^x$,$e$,and $\pi$
A numerical calculation on Mathematica shows that
$$I_1=\int_0^1 x^x(1-x)^{1-x}\sin\pi x\,\mathrm dx\approx0.355822$$
and
$$I_2=\int_0^1 x^{-x}(1-x)^{x-1}\sin\pi x\,\mathrm dx\approx1.15573$$
A ...
44
votes
3answers
3k views
$\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals.
I am aware of the calculation ...
43
votes
6answers
3k views
Ways to evaluate $\int \sec \theta \, d \theta$
The standard approach for showing $\int \sec \theta \, d \theta = \ln |\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a ...
36
votes
5answers
600 views
What is so special about $\alpha=-1$ in the integral of $x^\alpha$?
Of course, it is easy to see, that the integral (or the
antiderivative) of $f(x) = 1/x$ is $\log(|x|)$ and of course for
$\alpha\neq 1$ the antiderivative of $f(x) = x^\alpha$ is
...
35
votes
13answers
4k views
Proving $\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$
How to prove
$$\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$$
30
votes
5answers
1k views
Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$
I've got troubles in computing the below integral:
$$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$$
I hope it can be expressed in elementary functions. I've tried simple substitution as $u=\sin(x)$ ...
30
votes
4answers
820 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 ...
29
votes
3answers
688 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 ...
28
votes
5answers
1k views
Why are gauge integrals not more popular?
A recent answer reminded me of the gauge integral, which you can read about here.
It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue ...
28
votes
2answers
898 views
Evaluating $\int P(\sin x, \cos x) \text{d}x$
Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$.
For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$.
Is there a general method which allows us to evaluate the ...
27
votes
2answers
634 views
Making trigonometric substitutions rigorous
I've been tutoring some basic calculus, and it made me think about something pretty basic.
Let me explain the problem by example:
Say we are given the integral $\int \frac{x^2}{\sqrt{1-x^2}}dx$. It ...
26
votes
4answers
1k views
How do I integrate the following? $\int{\frac{(1+x^{2})\mathrm dx}{(1-x^{2})\sqrt{1+x^{4}}}}$
$$\int{\frac{1+x^2}{(1-x^2)\sqrt{1+x^4}}}\mathrm dx$$
This was a Calc 2 problem for extra credit (we have done hyperbolic trig functions too, if that helps) and I didn't get it (don't think anyone ...
26
votes
13answers
1k views
List of interesting integrals for early calculus students
I am teaching Calc 1 right now and I want to give my students more interesting examples of integrals. By interesting, I mean ones that are challenging, not as straightforward (though not extremely ...
26
votes
4answers
624 views
Generalizing the trick for integrating $\int_{-\infty}^\infty e^{-x^2}dx$?
There is a well-known trick for integrating $\int_{-\infty}^\infty e^{-x^2}dx$, which is to write it as $\sqrt{\int_{-\infty}^\infty e^{-x^2}dx\int_{-\infty}^\infty e^{-y^2}dy}$, which can then be ...
24
votes
4answers
435 views
How to prove that $\int_0^1\left(\sum_{k=n}^\infty {x^k\over k}\right)^2\,dx = \int_0^1 2x^{n-1}\log\left(1+{1\over\sqrt{x}}\right)\,dx.$
American Mathematical Monthly problem 11611 essentially asks you to show that
$$\lim_n\ n \int_0^1\left(\sum_{k=n}^\infty {x^k\over k}\right)^2\,dx=2\log(2).\tag1$$
This would follow easily from ...
23
votes
4answers
2k views
Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$
It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why?
I got:
$2\mathrm dy\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\mathrm ...
23
votes
2answers
631 views
On calculating $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ — where is the mistake?
I've seen the integral $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ on a thread in this forum and I tried to calculate it by using power series. I wrote the integral as a sum then again as an integral. ...
22
votes
8answers
1k views
When two functions are equal, but not.
I haven't looked into it much, but this is something I've been aware of that I know I need to look into.
When I have a function $f(x)=\frac{x+1}{x+1}$, There is a discontinuity at $x=-1$, yet ...
22
votes
2answers
495 views
Proving $ \int_{0}^{\infty} \frac{\ln(t)}{\sqrt{t}}e^{-t} \mathrm dt=-\sqrt{\pi}(\gamma+\ln{4})$
I would like to prove that:
$$ \int_{0}^{\infty} \frac{\ln(t)}{\sqrt{t}}e^{-t} \mathrm dt=-\sqrt{\pi}(\gamma+\ln{4})$$
I tried to use the integral $$\int_{0}^{n} ...
21
votes
4answers
921 views
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx$
Compute
$$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx$$
21
votes
7answers
673 views
Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm{dx}$
Evaluate
$$\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm{dx}$$
21
votes
7answers
992 views
Why is $\int_{0}^{\infty} \frac {\ln x}{1+x^2} \mathrm{d}x =0$?
We had our final exam yesterday and one of the questions was to find out the value of:
$$\int_{0}^{\infty} \frac {\ln x}{1+x^2} \mathrm{d}x $$
Interestingly enough, using the substitution ...
21
votes
4answers
918 views
Maybe a rather famous integral
How to evaluate :
$$\int_0^{\frac{\pi}{2}}\left(\frac{x}{\sin x}\right)^2\text{d}x$$
Thx guys! I was wondering how would use a series expansion?
21
votes
3answers
911 views
Integrate square of the log-sine integral: $\int_0^{\frac{\pi}{2}}\ln^{2}(\sin(x))dx$
$\displaystyle \int_{0}^{\frac{\pi}{2}} \ln(\sin(x))dx=-\frac{\pi}{2}\ln(2)$ is an integral that is common.
But, how can we show ...
21
votes
1answer
475 views
How to show that $\int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx = 0$
Evaluate the integral: $$ \int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx$$
The answer is $0,$ but I am unable to get it. There is some symmetry I can not see.
21
votes
4answers
502 views
The Meaning of the Fundamental Theorem of Calculus
I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
21
votes
3answers
843 views
The right “weigh” to do integrals
Back in the day, before approximation methods like splines became vogue in my line of work, one way of computing the area under an empirically drawn curve was to painstakingly sketch it on a piece of ...
21
votes
3answers
296 views
+500
Conjectural closed-form representations of sums, products or integrals
What are some examples of infinite sums, products or definite integrals that have conjectural closed form representations that were confirmed by numerical calculations up to whatever maximum precision ...
20
votes
4answers
772 views
Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$?
Let $a$ be a non-zero real number.
Is it true that $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ is independent on $a$ ?
Any proof?
20
votes
6answers
960 views
What's the nth integral of $\frac1{x}$?
It can be shown by simple induction that $\dfrac{\mathrm d^n}{\mathrm dx^n}\left(\dfrac1{x}\right) = \dfrac{(-1)^n n!}{x^{n+1}}$.
But what about the nth integral of $\dfrac1{x}$? Finding the first ...
20
votes
7answers
519 views
How to evaluate $I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)dx$
Find the value of
$I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)dx$
We have the information that
$J=\displaystyle\int_0^{\pi/2}x\ln(\sin x)\ln(\cos ...
20
votes
4answers
1k views
When is an elliptic integral expressible in terms of elementary functions?
After seeing this recent question asking how to calculate the following integral
$$ \int \frac{1 + x^2}{(1 - x^2) \sqrt{1 + x^4}} \, dx $$
and some of the comments that suggested that it was an ...
20
votes
2answers
199 views
$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$
I need to find a closed-form for the following integral. Please give me some ideas how to approach it:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
20
votes
4answers
820 views
What is to geometric mean as integration is to arithmetic mean?
The arithmetic mean of $y_i ... y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$
For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and using ...
20
votes
3answers
884 views
tough integral involving $\sin(x^2)$ and $\sinh^2 (x)$
I ran across this integral I get no where with. Can someone suggest a method of attack?.
$$\int_0^{\infty}\frac{\sin(\pi x^2)}{\sinh^2 (\pi x)}\mathrm dx=\frac{2-\sqrt{2}}{4}$$
I tried series, ...
20
votes
1answer
451 views
Integral for function of square
Let $f:[0,\infty) \rightarrow \mathbb R$ be a strictly positive, decreasing, differentiable function, such that
$$f(0) = 1, \quad \lim_{x\rightarrow \infty} f(x) = 0$$
and
$$\frac{1}{f(x)^2} = ...
19
votes
2answers
480 views
$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$ Evaluate Integral
Evaluate
$$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$$
19
votes
2answers
703 views
Tricky contour integral
I am trying to evaluate the definite integral
$$\int_0^\infty \frac{\sin ax\ dx}{x^2+b^2}$$
where $a,b>0$. This is a problem on an assignment for a class in complex variables. I understand the ...
19
votes
1answer
393 views
If two definite integrals are equal, does there exist a chain of substitutions and/or partial integrations which will get us from one to the other?
Earlier today I was having a little fun with Catalan's constant and its various integral representations: showing that they all do indeed evaluate to the same thing. This got me wondering whether this ...
18
votes
5answers
698 views
Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
I have seen the Fresnel integral
$$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$
evaluated by contour integration and other complex analysis methods, and I have found these methods to be the ...
18
votes
3answers
239 views
+100
An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$
I need to calculate the following integral:
$$\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$$
where
$$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$
...
18
votes
1answer
512 views
Interesting integral formula
I looked around and found that integrals of the form
$$\int_{0}^{\infty} \frac{x^{m-1}}{a+x^n}, a,m,n \in \mathbb{R}, 0<m<n, 0<a$$
seem to occur very often:
Just to give a few examples ...
18
votes
2answers
624 views
Does integration by parts with “deja vu” have a name?
In some integration by parts problems, such as evaluating the integral of $e^x \cos x$ or $\sec^ 3 x$, one performs integration by parts (possibly more than once, and possibly together with algebraic ...
17
votes
5answers
2k views
Integral of $\frac{1}{(1+x^2)^2}$
I am in the middle of a problem and having trouble integrating the following integral:
$$\int_{-1}^1\frac1{(1+x^2)^2}\mathrm dx$$
I tried doing partial fractions and got:
$$1=A(1+x^2)+B(1+x^2)$$
I ...
17
votes
3answers
7k views
Why is the area under a curve the integral?
I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself.
However when it comes to the area ...
17
votes
4answers
575 views
Evaluation of $\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)$?
I would like to evaluate the sum
$$
\sum\limits_{n=0}^\infty \left(\operatorname{Si}(n)-\frac{\pi}{2}\right)
$$
Where $\operatorname{Si}$ is the sine integral, defined as:
$$\operatorname{Si}(x) := ...
17
votes
2answers
463 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?.
...
