Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
7,559
questions
109
votes
11
answers
9k
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Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$
I've been looking at
$$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$
It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:
$$\...
- 119k
348
votes
7
answers
48k
views
How can you prove that a function has no closed form integral?
In the past, I've come across statements along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations:
addition/...
- 5,431
210
votes
19
answers
144k
views
Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$
How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
- 7,768
21
votes
3
answers
3k
views
Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\,\mathrm dx$. [duplicate]
Evaluate the integral $$\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}\, \mathrm dx.$$
How can i evaluate this one? Didn't find any clever substitute and integration by parts doesn't lead ...
- 2,443
39
votes
2
answers
9k
views
Fourier series of Log sine and Log cos
I saw the two identities
$$
-\log(\sin(x))=\sum_{k=1}^\infty\frac{\cos(2kx)}{k}+\log(2)
$$
and
$$
-\log(\cos(x))=\sum_{k=1}^\infty(-1)^k\frac{\cos(2kx)}{k}+\log(2)
$$
here: twist on classic log of ...
- 2,132
116
votes
9
answers
14k
views
Proof of Frullani's theorem
How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are
Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously ...
- 3,455
111
votes
12
answers
15k
views
Calculating the integral $\int_0^\infty \frac{\cos x}{1+x^2}\, \mathrm{d}x$ without using complex analysis
Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form?
$$\int_0^\infty\frac{\cos x}{1+x^2}\,\mathrm{d}x$$
- 6,678
45
votes
11
answers
21k
views
Computing the integral of $\log(\sin x)$
How to compute the following integral?
$$\int\log(\sin x)\,dx$$
Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find ...
- 495
66
votes
8
answers
102k
views
Finding $\int x^xdx$
I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this:
Let $u=x^x$.
$$\begin{align}
\int x^x \, dx&=\int u \, du\\[6pt]
&=\frac{u^2}{2}\\[6pt]
&=\dfrac{\left(x^...
- 2,238
54
votes
3
answers
3k
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 ...
88
votes
15
answers
27k
views
Proof of $\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$
I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 \mathrm dx=\frac{\pi}{2}.$$
What do you think?
It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} \mathrm ...
- 14k
169
votes
8
answers
41k
views
Lesser-known integration tricks
I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now ...
59
votes
3
answers
6k
views
Areas versus volumes of revolution: why does the area require approximation by a cone?
Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$. Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder. ...
- 18.6k
16
votes
1
answer
4k
views
Getting different answers when integrating using different techniques
Question: Is it possible to get multiple correct results when evaluating an indefinite integral? If I use two different techniques to evaluate an integral, and I get two different answers, have I ...
- 25.7k
102
votes
8
answers
43k
views
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$
Compute
$$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
- 43.7k
30
votes
9
answers
33k
views
Evaluate $\int_0^{\pi/2}\log\cos(x)\,\mathrm{d}x$
How can you evaluate $$\int\limits_0^{\pi/2}\log\cos(x)\,\mathrm{d}x\;?$$
- 353
33
votes
7
answers
14k
views
Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$
For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$.
My attempt: Let
$$\begin{align*}
f_n(x)
&= \frac{\ln\left(1-2
\left(a+\frac{1}...
- 1,696
56
votes
6
answers
31k
views
Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral?
In this wikipedia article for improper integrals,
$$
\int_0^{\infty}\frac{\sin x}{x}dx
$$
is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) ...
user9464
38
votes
5
answers
5k
views
Integrate $\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$
Evaluate the integral $$\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x$$
22
votes
4
answers
2k
views
Frullani 's theorem in a complex context.
It is possible to prove that $$\int_{0}^{\infty}\frac{e^{-ix}-e^{-x}}{x}dx=-i\frac{\pi}{2}$$ and in this case the Frullani's theorem does not hold since, if we consider the function $f(x)=e^{-x}$, we ...
- 32.4k
9
votes
5
answers
1k
views
A group of important generating functions involving harmonic number.
How to prove the following identities:
$$\small{\sum_{n=1}^\infty\frac{H_{n}}{n^2}x^{n}=\operatorname{Li}_3(x)-\operatorname{Li}_3(1-x)+\ln(1-x)\operatorname{Li}_2(1-x)+\frac12\ln x\ln^2(1-x)+\zeta(3)...
- 23.4k
36
votes
5
answers
13k
views
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem ...
- 4,325
24
votes
4
answers
3k
views
What is wrong with the following u-substitution?
We will calculate $\displaystyle\int^{2 \pi}_0 x \, dx$. Let $u=\sin (x)$, and observe that $\sin(2 \pi)=0$ and $\sin(0)=0$. We also have that $\frac{du}{dx}=\cos(x)=\sqrt{1-u^2}$. Hence,
$$
\int^{2 ...
- 241
77
votes
7
answers
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views
How to prove $\int_{-\infty}^{+\infty} f(x)dx = \int_{-\infty}^{+\infty} f\left(x - \frac{1}{x}\right)dx?$
If $f(x)$ is a continuous function on $(-\infty, +\infty)$ and $\int_{-\infty}^{+\infty} f(x) \, dx$ exists. How can I prove that
$$\int_{-\infty}^{+\infty} f(x) \, dx = \int_{-\infty}^{+\infty} f\...
- 1,771
11
votes
5
answers
1k
views
Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$
Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to
\infty} s_n$.
\begin{align}
s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx
\end{align}
I have ...
user14082
75
votes
9
answers
12k
views
Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods?
I have seen the Fresnel integral
$$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$
evaluated by contour integration and other complex analysis methods, and I have found these methods to be the ...
- 24.7k
20
votes
8
answers
3k
views
Indefinite integral of secant cubed $\int \sec^3 x\>dx$
I need to calculate the following indefinite integral:
$$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)$$
but I don't ...
- 6,211
8
votes
2
answers
2k
views
What are BesselJ functions?
I solved an integration on mathematica which gives BesselJ functions and some other terms. I explored mathematica help and google but could not understand the difference between different types of ...
- 135
132
votes
6
answers
49k
views
Why is integration so much harder than differentiation?
If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product ...
- 1,501
106
votes
14
answers
11k
views
Ways to evaluate $\int \sec \theta \, \mathrm d \theta$
The standard approach for showing $\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$ is to multiply by $\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then ...
- 54.1k
37
votes
7
answers
11k
views
Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
user55225
50
votes
11
answers
40k
views
Why do we require radians in calculus?
I think this is just something I've grown used to but can't remember any proof.
When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
- 9,530
47
votes
4
answers
9k
views
Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$
It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why?
I got:
$2\,\mathrm dy\,\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\,\...
- 5,379
39
votes
8
answers
4k
views
Simpler way to compute a definite integral without resorting to partial fractions?
I found the method of partial fractions very laborious to solve this definite integral :
$$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$
Is there a simpler way to do this ?
- 829
25
votes
7
answers
25k
views
Gaussian-like integral : $\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$
Prove that :
$$ \frac{\sqrt{\pi}}{2} e^{-\frac{a^2}{4} } =\int_0^{\infty} e^{-x^2} \cos( a x) \ \mathrm{d}x$$
the only thing I can think of is differentiating the RHS and trying to get :
$$ -2 f'(a)...
- 4,581
4
votes
1
answer
1k
views
A few improper integral
$$\displaystyle \begin{align*}
& \int_{0}^{+\infty }{\frac{\text{d}x}{1+{{x}^{n}}}} \\
& \int_{-\infty }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x} \\
& \int_{0}^{+\infty }{...
- 3,815
62
votes
7
answers
68k
views
Evaluating the indefinite integral $ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $ [closed]
I have been having extreme difficulties with this integral. I would appreciate any and all help.
$$
\int \sqrt{\tan x} ~ \mathrm{d}{x}.
$$
- 2,029
167
votes
20
answers
28k
views
Striking applications of integration by parts
What are your favorite applications of integration by parts?
(The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!)
Thanks for your ...
56
votes
7
answers
9k
views
What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$?
/A problem from the 2012 MIT Integration Bee is
$$
\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx
$$
The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral ...
- 571
44
votes
5
answers
3k
views
Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only
I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\in \...
- 1,438
37
votes
3
answers
10k
views
$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$ [closed]
Trying to solve
$f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges.
I need to prove that:
$$\lim \limits_{x \to \infty} f(x) = 0$$
Would ...
- 637
266
votes
5
answers
22k
views
Evaluate $ \int_{0}^{\frac{\pi}2}\frac1{(1+x^2)(1+\tan x)}\:\mathrm dx$
Evaluate the following integral
$$
\tag1\int_{0}^{\frac{\pi}{2}}\frac1{(1+x^2)(1+\tan x)}\,\mathrm dx
$$
My Attempt:
Letting $x=\frac{\pi}{2}-x$ and using the property that
$$
\int_{0}^{a}f(x)\,\...
- 50.7k
24
votes
3
answers
4k
views
Series as an integral (sophomore's dream)
I need help with this exercise.
I need to prove
$$\int_{0}^{1}x^{-x}\ dx=\sum_{n=1}^{\infty}n^{-n}$$
I think I should use some convergence theorem, but I'm stuck.
Thanks a lot!
- 243
85
votes
3
answers
102k
views
Can a limit of an integral be moved inside the integral?
After coming across this question: How to verify this limit, I have the following question:
When taking the limit of an integral, is it valid to move the limit inside the integral, providing the ...
- 10.1k
70
votes
5
answers
3k
views
Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$
Show that :
$$
\int_{0}^{\Large\frac\pi2}
{\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right)
\ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right)
\over
\cos\left(x\right)\sin\left(x\...
- 3,815
102
votes
6
answers
123k
views
Why is the area under a curve the integral?
I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself.
However when it comes to the area ...
- 1,277
28
votes
3
answers
22k
views
How to integrate $ \int \frac{x}{\sqrt{x^4+10x^2-96x-71}}dx$?
I read about $ \int \dfrac{x}{\sqrt{x^4+10x^2-96x-71}}dx$ on the Wikipedia Risch algorithm page. They gave an answer but I don't understand how they got it.
- 1,688
20
votes
7
answers
2k
views
How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$?
I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it ...
- 10.1k
19
votes
2
answers
32k
views
Prove $\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$
I am trying to prove $$\int\cos^n x \ dx = \frac{1}n \cos^{n-1}x \sin x + \frac{n-1}{n}\int\cos^{n-2} x \ dx$$
This problem is a classic, but I seem to be missing one step or the understanding of ...
- 4,597
8
votes
6
answers
8k
views
Integrate $e^{ax}\sin(bx)?$
Is there a general formula for finding the primitive of $$e^{ax}\sin(bx)?$$
I've done this manually with $a=9$ and $b=4$ using Euler's formulas. But it takes a bit of time. Is there a pattern here?
- 2,705