All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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1answer
113 views

Normalization of a two-dimensional kernel function

I've got three two-dimensional kernel functions which look like this $$ k(r,h) = n \cdot \begin{cases} \ldots & 0 \le r \le h \\ 0 & otherwise \end{cases} $$ With ...
7
votes
2answers
221 views

Proving that $\int_0^1\frac{x \log^2(1-x)}{1+x^2} \ dx = \frac{35}{32}\zeta(3)+\frac{1}{24}\log^3(2) -\frac{5}{96} \pi^2 \log(2)$

Could we possibly prove this result without using the polylogarithm? I know how to do it by polylogarithm means, but I want a different way. Is that possible? $$\int_0^1\frac{x \log^2(1-x)}{1+x^2} ...
2
votes
3answers
130 views

Integration - finding an explicit formula

The question in my textbook asks: If $f$ is a continuous function such that $$\int\limits_0^x{f(t)dt}=xe^{2x}+\int\limits_0^x{e^{-t}f(t)dt}$$ for all $x$, find an explicit formula for $f(x)$. My ...
2
votes
4answers
114 views

How to prove $\int_0^\pi \frac{dx}{2+2\sin x+\cos x}=\log3$?

How can we prove that: $$\int_0^\pi \frac{dx}{2+2\sin x+\cos x}=\log3$$ I don't have any ideas, the $f(\pi-x)$ thing doesn't work as well. Please help :)
1
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2answers
43 views

Doubt in integral substitution

I am not able to figure out what substitution to use in the following integral $$ \int \frac{(x-1)e^x}{(x+1)^3}dx $$ Any help would be appreciated.
2
votes
2answers
50 views

Examples of interesting integrable functions with at least 2 fixed points and an explicit inverse

What are some interesting functions I can use to demonstrate this integration trick: $$\int_a^b [f(x)+f^{-1}(x)]=b^2-a^2$$ I would like to know of some interesting functions where this trick is not ...
1
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2answers
67 views

Triple Integral exercise

Calculate $\int\int\int_Dz\;dxdydz$ if $D$ is the region inside $z=0,z=\sqrt{x^2+y^2}$ and $x^2+y^2=1$. I would like to know if the answer I got is right. This is what I did: $(1)$ Change to ...
6
votes
1answer
69 views

Integral involving square root of sine and cosine

Is there any closed formula for $$ \int_{0}^{\pi/2} \dfrac{e^{-x}\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin x}} $$ I know $$ \int_{0}^{\pi/2} \dfrac{\sqrt{\cos x}\ dx}{\sqrt{\cos x} + \sqrt{\sin ...
4
votes
1answer
89 views

Prove this polylogarithmic integral has the stated closed form value

Question. Prove the following polylogarithmic integral has the stated value: $$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$ ...
3
votes
1answer
48 views

Does $\int { y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$ have a closed form

I am trying to solve the following indefinite integral $$F_Y(y) = \int {y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$$ Where $J_0$ is the Bessel function of the first kind. I tried ...
2
votes
1answer
80 views

Inconventional Integral inequality

$$\int_a^bw(x)|f(x)||g(x)|\;dx \le \left(\int_a^bw(x)\;dx\right) \max_{a\le x\le b}|f(x)|\cdot \max_{a\le x\le b}|g(x)|$$ I don't really understand this integral inequality. How do I go about ...
2
votes
3answers
109 views

Evaluate $\int x e^{\sqrt{x}} \, dx$

$$\int_0^1 xe^{\sqrt{x}} dx = ? $$ All I can think of is the integration by parts rule, where $ u = x $ and $ dv= e^{\sqrt(x)} $ $ \Rightarrow du = 1$ and $ v= e^{\sqrt(x)} $ The answer I get is ...
0
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2answers
56 views

How can the double intergal expression be reduced to the single intergal expression

Consider the following expression where $x(s)$ and $y(s)$ are continuous as is necessary on the closed interval [a,b]. (This is a functional analysis question -- see below for details.) $$x(s) = ...
11
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3answers
225 views

Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...
12
votes
1answer
146 views

$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$ [duplicate]

$$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$$ I tried Integration by parts but failed. Wolfram alpha gives answer in decimal points which are same as of $2\pi$. Any hints or suggestions will be helpful.
0
votes
2answers
67 views

Integrate $\int \sin^4x \cos^2x dx$

Integrate $$\int \sin^4x \cos^2x dx$$ Now, there's few solutions to this problem already on the internet. For example on yahoo: https://answers.yahoo.com/question/index?qid=20090204203206AAbjUfM and ...
4
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0answers
52 views

Clarification of Contour Integration [duplicate]

I apologise if this seems like an elementary and silly question, but I am confused about the integral $$I=\int^{\infty}_{-\infty}\frac{\cos{x}}{1+x^2}dx=\frac{\pi}{e}$$ If I consider a semicircular ...
3
votes
5answers
193 views

Calculating the area

For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them; Attempt to solve: Limits of the integral are $1$ and $-3$, so I took ...
2
votes
1answer
87 views

Improper integral $\int_{0}^{\pi} \frac{x}{\sin x} dx$

Find out whether or not the following integral exists $$\int_{0}^{\pi} \frac{x}{\sin x} dx.$$ I'm pretty sure this integral doesn't exist but I can't seem to find a good way to prove this. It ...
1
vote
3answers
92 views

Evaluate $\int \frac{1}{(2x+1)\sqrt {x^2+7}}dx$

How to do this indefinite integral (anti-derivative)? $$I=\displaystyle\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}dx$$ I tried doing some substitutions ($x^2+7=t^2$, $2x+1=t$, etc.) but it didn't work out.
0
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0answers
36 views

The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
1
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0answers
63 views

Integration Error

Sorry if this doesn't make any sense or if I did something obviously wrong, I was just playing around with taylor series' and then I got stuck. I know from the geometric series that: ...
1
vote
2answers
41 views

Euler's method for first three approximations?

I have tried variations of the problem for an hour at least and cannot get around to sloving this one. Thank you for input!
3
votes
1answer
34 views

Volume of a solid(between two planes)?

A solid lies between planes perpendicular to the y-axis at $ y=0$ and $y=1$. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola ...
1
vote
1answer
38 views

Integration: substitution then differentiation result different to differentiation then substitution.

I want to simplify this derivative ($n$ is an integer) $$ \frac{d}{d\theta} \int_{0}^{2\pi} e^{i n \phi} e^{i 2\pi k r \cos(\phi - \theta)} d\phi $$ If I substitute $\psi = \phi - \theta$ and ...
1
vote
1answer
32 views

Length of a curve?

I know how to find arc length and set up the equation in normal circumstances, but I have failed in all attempts to even set up this problem. I cannot even find a good example similar to this to get ...
2
votes
2answers
39 views

Shell method to find the volume of a solid?

Region bounded by $y=3x-2$, $y=\sqrt{x}$, and $x=0$ about the $y$-axis. I have been doing the washer method for all of my problems up to this one, and cannot seem to find a good resource to help guide ...
0
votes
1answer
36 views

Line integral calculation

Problem Let $f:[-1,1] \to \mathbb R$ be a $C^1$ function such that $f(-1)=0=f(1)$ and $f>0$ on $(-1,1)$. Knowing that the graph of $f$ is contained in the set $\{(x,y) : x^2+y^2\leq 1, y\geq 0 ...
5
votes
1answer
135 views

Hard integral, low hints… [duplicate]

$$\int_{ - \pi /2}^{\pi /2} \frac1{2007^{x} + 1}\cdot \frac {\sin^{2008}x}{\sin^{2008}x + \cos^{2008}x} \, dx .$$ This integral stuns me for a while, I just can't solve it! I tried integration by ...
5
votes
2answers
99 views

Is $\int^x \cos \frac1t$ differentiable at zero?

From Spivak's Calculus, 4th ed., exc 14-20: Let $$f(x) = \begin{cases} \cos \frac1x, & x\neq 0\\ 0, &x=0. \end{cases}$$ Is the function $\int_0^xf$ differentiable at zero? I'm having ...
0
votes
2answers
63 views

How to prove $ \int \sqrt{a^2-u^2}du $

How can I prove that the following definite integral? $$ \int \sqrt{a^2-u^2}du = \frac{1}{2}\left[u\sqrt{a^2 - u^2} + a^2 \arcsin\left(\frac{u}{a}\right)\right] +C$$
0
votes
1answer
22 views

Not getting limits of integration right (triple integrals)?

Calculate the volume bounded by $y+z=1$, $z=x^2-1$, $z=1-x^2$ and $y=0$ So our volume $V=\int \int \int _D1 dA$, all that's left is figuring out limits of integration. One method I saw was looking at ...
0
votes
1answer
70 views

Show that $\int_{\pi/4}^{\pi/2} \frac{\sin x}{x}\,dx\leq \frac{\sqrt{2}}{2}$

Show that $$\int_{\pi/4}^{\pi/2} \dfrac{\sin x}{x}\,dx\leq \dfrac{\sqrt{2}}{2}$$ Any Ideas, how to start ?!
-1
votes
1answer
37 views

Proving the indefinite integral $ \int \frac{1}{u^2(a+bu)}du $ [closed]

How can I prove that the indefinite integral $$ \int \frac{1}{u^2(a+bu)}du $$ is equal to $$ -\frac{1}{a}\left(\frac{1}{u}+\frac{b}{a}\ln\left|\frac{u}{a+bu}\right|\right)+C\ ? $$
2
votes
4answers
135 views

What is the integral of x/ln(x)?

Well, I'm french so excuse me if I make some mistakes in english... I have to calculate this integral : $$ \int_{e}^{2e} \frac{x}{\ln(x)} dx $$ But I don't know how, can you help me please? Thank ...
0
votes
1answer
27 views

Euler's method and Riemann sum

For: $F(0) = 0$ and $F'(x) = f(x)$ Euler's method: $F(0+h) = F(0)+ hF'(0) = 0 + hf(0)$ Continuing the process, $F(10h) = hf(0)+hf(h)+hf(2h)+.....hf(9h)$ This resembles the Riemann sum: ...
5
votes
0answers
131 views

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
5
votes
1answer
49 views

Maximum value problem

A function $\hspace{0.1cm}$$f:[0,1]\to[-1,1]$$\hspace{0.1cm}$ satisfying$\hspace{0.1cm}$ $|f(x)|\leq x$$\hspace{0.1cm}$ $\forall x\in[0,1]$. Then find the maximum value of: ...
0
votes
1answer
27 views

Newton-Cotes Quadrature formula

Im trying to find more information about numerical integration methods. When is a Newton-Cotes Quadrature formula on n nodes exact?
2
votes
1answer
36 views

Upper bound the integral or its PV (or prove that it diverges)

I need help in finding an upper bound the following integral (or its Cauchy Principal Value): $$ \int_0^1 \sqrt{x} \frac{|\ln(\frac{(y^{-1}-1)}{(x^{-1}-1)})|}{|x-y|} dx $$ This integral arises as ...
0
votes
1answer
20 views

Finding the boundaries of integration when calculating P(X + Y > a) or P(X + Y < b) (Jointly Distributed Continuous Random Variables)

I have a problem on setting the boundaries of integration when I'm trying to find probabilities like $P(X + Y > a)$ or $P(X + Y < b)$. For example, when I have $f(x,y) = \frac {x} {5}\ +\frac ...
6
votes
3answers
249 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
4
votes
4answers
111 views

If $f(x)$ is discontinuous at $x=0$, can $\int_{-1}^1 f(x)dx$ exist.

I am interested in the reasoning. All help is appreciated
-1
votes
2answers
69 views

Prove that $\displaystyle\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$

Prove that $\displaystyle\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$ Note that $c$ need not belong to $(a, b)$ And $f(x)$ is a continuous function. All ideas are appreciated.
1
vote
1answer
21 views

Double integral calculation where $x=(y-1)^{2}-1$ and $y=x$. Not sure whether I should do it in terms of $y$ or $x$?

This is what it looks like: My first strategy was to separate it into two by drawing a vertical line at x=0 and calculate the first half in terms of x first, and the second half in terms of y ...
8
votes
2answers
214 views

Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of ...
1
vote
1answer
74 views
5
votes
3answers
159 views

Examples of “difficult” integrals with are easier to solve with a series?

Yesterday someone posted an extremely elegant solution to a seemingly bizarre series where the integral: $$\int_{0}^{1} x^{m}\ dx = \frac{1}{m + 1}$$ was utilized. Oftentimes one will also ...
7
votes
1answer
136 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
5
votes
3answers
110 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...