Questions on the evaluation of definite and indefinite integrals
34
votes
13answers
4k views
Proving $\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$
How to prove
$$\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$$
10
votes
4answers
707 views
Simpler way to compute a definite integral without resorting to partial fractions?
I found the method of partial fractions very laborious to solve this definite integral :
$$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$
Is there a simpler way to do this ?
28
votes
2answers
897 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 ...
11
votes
1answer
1k views
Will moving differentiation from inside, to outside an integral, change the result?
I'm interested in the potential of such a technique. I got the idea from Moron's answer to this question, which uses the technique of differentiation under the integral.
Now, I'd like to consider ...
16
votes
7answers
3k views
Proof for an integral involving sinc function
I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\frac{\pi}{2}.$$
What do you think?
It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} dx$$ is also ...
16
votes
4answers
1k views
Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis
Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form?
$$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
10
votes
5answers
1k views
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem ...
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 ...
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 ...
6
votes
2answers
283 views
Methods to evaluate $ \int _{a }^{b }\!{\frac {\ln \left( tx + u \right) }{m{x}^{2}+nx +p}}{dx} $
Today I saw a question with an answer that made me rethink of the following question, since it's not the first time I try to find an answer to it. If you look at the answer of Mhenni Benghorbal
here ...
40
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 ...
12
votes
4answers
2k views
Prove: $\int_{0}^{\infty} \sin (x^2) dx$ converges.
$\sin(x^2)$ is an example for a function which its limit when $x \to \infty$ is not $0$, and still its integral from $0$ to $\infty$ is finite. I'd like your help with understanding why and a ...
9
votes
5answers
701 views
Help solving $\int {\frac{8x^4+15x^3+16x^2+22x+4}{x(x+1)^2(x^2+2)}dx}$
$\displaystyle\int {\frac{8x^4+15x^3+16x^2+22x+4}{x(x+1)^2(x^2+2)}\,\mathrm{d}x}$
I used partial fractions, solved $A = 2, C = 3$.
$$\frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2} ...
17
votes
3answers
7k views
Why is the area under a curve the integral?
I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself.
However when it comes to the area ...
8
votes
3answers
327 views
A log improper integral
Evaluate :
$$\int_0^{\frac{\pi}{2}}\ln ^2\left(\cos ^2x\right)\text{d}x$$
I found it can be simplified to
$$\int_0^{\frac{\pi}{2}}4\ln ^2\left(\cos x\right)\text{d}x$$
I found the exact value in the ...
12
votes
3answers
534 views
$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$
Trying to solve
$f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges.
I need to prove that:
$$\lim \limits_{x \to \infty} f(x) = 0$$
Would ...
7
votes
4answers
329 views
Evaluating $\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta$
I need solve this integral, and I tried various methods of solving and did not get it. The integral is:
$$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta,$$
where $t$ is a ...
18
votes
5answers
693 views
Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
I have seen the Fresnel integral
$$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$
evaluated by contour integration and other complex analysis methods, and I have found these methods to be the ...
21
votes
4answers
906 views
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx$
Compute
$$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} dx$$
11
votes
2answers
413 views
Frullani proof integrals
How can i prove the theorem of Frullani? I did not even know all the hypothesis that f must satisfy, but I think that this are
Let $f:\left[ {0,\infty } \right] \to \mathbb R$ be a a continuously ...
6
votes
1answer
380 views
Summing over General Functions of Primes and an Application to Prime $\zeta$ Function
Along the lines of thought given here, is it in general possible to substitute a summation over a function $f$ of primes like the following:
$$
\sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t))\tag{1}
$$
and ...
4
votes
3answers
190 views
Indefinite integral of secant cubed
I need to calculate the following indefinite integral:
$$I=\int \frac{1}{\cos^3(x)}dx$$
I know what the result is (from Mathematica):
$$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$
but I don't ...
2
votes
2answers
725 views
How to integrate $\int e^{-t^{2}} \space dt $ using introductory calculus methods
Earlier today I stumbled across this when I was doing some practice questions for a physics course: $$\int e^{-t^2} \space dt $$
To expand, the limits of integration were something like $1$ and $4$ ...
1
vote
6answers
484 views
Integral of $\int e^{2x} \sin 3x\, dx$
I am suppose to use integration by parts but I have no idea what to do for this problem
$$\int e^{2x} \sin3x dx$$
$u = \sin3x dx$ $du = 3\cos3x$
$dv = e^{2x} $ $ v = \frac{ e^{2x}}{2}$
From this I ...
13
votes
5answers
797 views
Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$
I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general:
$$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$
...
14
votes
4answers
545 views
Evaluation of the integral $\int_0^1 \frac{\ln(1 - x)}{1 + x}dx$
How can I evaluate the integral
$$\int_0^1 \frac{\ln(1 - x)}{1 + x}dx$$
I tried manipulating the known integral
$$\int_0^1 \frac{\ln(1 - x)}{x}dx = -\frac{\pi^2}{6}$$
but couldn't do anything with ...
7
votes
4answers
572 views
How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$?
I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it ...
12
votes
4answers
1k views
Computing the integral of $\log (\sin x)$
How to compute the following integral
$$\int \log(\sin x)\,\mathrm dx?$$
9
votes
2answers
1k views
integral with exponential function and logarithm
$$
\int_0^{\infty } \frac{\log (x)}{e^x+1} \, dx = -\frac{1}{2} \log ^2(2)
$$
Anyone an idea on how to prove this?
7
votes
3answers
734 views
Name of this identity? $\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$
Again:
$$\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha^2+\beta^2}$$
Also the one for $\sin$:
$$\int e^{\alpha x}\sin(\beta x) ...
2
votes
2answers
181 views
Complex Fourier series
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
21
votes
3answers
910 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 ...
19
votes
2answers
475 views
$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$ Evaluate Integral
Evaluate
$$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$$
13
votes
1answer
419 views
A nice log trig integral
Show that :
$$\int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\cos x{{\ln }^{2}}\sin x}{\cos x\sin x}}\text{d}x=\frac{1}{4}\left( 2\zeta \left( 5 \right)-\zeta \left( 2 \right)\zeta \left( 3 \right) ...
10
votes
2answers
184 views
Laplace's method
I'm still having a little trouble applying Laplace's method to find the leading asymptotic behavior of an integral. Could someone help me understand this? How about with an example, like:
...
6
votes
3answers
200 views
A generalized integral need help
I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$
What I do is use a Reciprocal subsitution, easy to show that:
...
10
votes
5answers
2k views
If $\int_0^x f \ dm$ is zero everywhere then $f$ is zero almost everywhere
I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some ...
1
vote
1answer
190 views
A few improper integral
$$\displaystyle \begin{align*}
& \int_{0}^{+\infty }{\frac{\text{d}x}{1+{{x}^{n}}}} \\
& \int_{-\infty }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x} \\
& \int_{0}^{+\infty ...
3
votes
1answer
112 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$
...
2
votes
3answers
67 views
how to solve an definite integral of floor valute function?
the question is, how to proof that this integral:
(the integral is definite from 0 to n^2)
$$\int_0^{n^2}\lfloor\sqrt{t}\rfloor dt = \frac{1}{6}n(n-1)(4n+1)$$
i'd very much appreciate your help on ...
2
votes
1answer
337 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 ...
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 ...
14
votes
7answers
2k views
Lebesgue integral basics
I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take ...
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 ...
29
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 $u=\sin(x)$ ...
14
votes
5answers
486 views
Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$
Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem.
Given hint: consider $f(z) = \ln ( 1 +z)$.
EDIT:: I know how to evaluate it, but I am ...
21
votes
1answer
471 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.
18
votes
1answer
509 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 ...
8
votes
4answers
631 views
Notation question: Integrating against a measure
Suppose $\mu$ is a measure. Is there any difference in meaning between the notation
$\int f(x)d\mu(x)$
and the notation
$\int f(x) \mu(dx)$?
Many thanks.
5
votes
4answers
318 views
Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
