Questions about the evaluation of specific definite integrals.

learn more… | top users | synonyms (1)

0
votes
0answers
5 views

Error estimate for Midpoint rule of ratio of integrals

Let's say that I partition an interval $[a,b]$ such that $x_{0} = a$, $x_{k} = a + k\Delta$, until $x_{K} = b$ $\Delta$ is the length of the subinterval. I assume equal length, and thus $\Delta = ...
2
votes
0answers
23 views

Derivation of Gradshteyn and Ryzhik integral 3.876.1 (in question)

In the Gradshteyn and Ryzhik Table of Integrals, the following integral appears (3.876.1, page 486 in the 8th edition): \begin{equation} \int_0^{\infty} \frac{\sin (p \sqrt{x^2 + a^2})}{\sqrt{x^2 + ...
0
votes
0answers
14 views

Volume of partially filled spherical cap? [on hold]

I have a spherical cap... the plane end (which is of course a circle) is vertically to ground... the radius of the sphere from it we made these cap is R, the distance from center of sphere to the ...
5
votes
4answers
95 views

Find $\int_0^1(\ln x)^n\hspace{1mm}dx$

I am not a big fan of induction, it's just a personal preference. Is there a method other than induction. Answer is $n!$ by the way
1
vote
0answers
24 views

Calculating in closed form $\int_0^{\infty} \frac{\text{PolyLog}^{(1,0)}(1,-x)}{1+x^2} \, dx$

Can you confirm the following result? Mathematica and other computational stuff I used seem unable to do anything about this result. Maybe to confirm it numerically? $$\int_0^{\infty} ...
2
votes
0answers
31 views

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
0
votes
0answers
72 views

Conjecturing the closed form $\frac{\pi ^2}{8}-\frac{\pi ^2}{8 \sqrt{2}}+\frac{\pi \log (2)}{4 \sqrt{2}}$

I conjecture that $$\small \int_0^{\pi/2} \frac{\cos ^2(x) \left(-2 \log \left(4^{-\sin ^2(x)} \sin ^{-4 \sin ^2(x)}(x)\right)-4 \log (\cos (x))+\cos (2 x) (4 \log (\cos (x))+\pi +\log ...
-4
votes
0answers
37 views

How to evaluate the integral $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$? [on hold]

How to evaluate the integral $\int^{1/2}_0\int^{1-x}_x (x+y)^9(x-y)^9 \, dy \, dx$?
1
vote
2answers
34 views

Volume of Solid Enclosed by an Equation

I'm having problems finding the triple integrals of equations. I guess it has to do with the geometry. Can someone solve the two questions below elaborately such that I can comprehend this triple ...
0
votes
0answers
25 views

Area under bijective decreasing function

Let $ f:[2,4]\to[3,5]$ be a bijective decreasing function,then find the value of $\int_{2}^{4}f(t) dt-\int_{3}^{5}f^{-1}(t) dt.$ I am not sure whether $\int_{2}^{4}f(t) dt=\int_{3}^{5}f^{-1}(t) dt$ ...
3
votes
1answer
25 views

Solve $4\int_0^a\int_0^{\sqrt{a^2-x^2}}\int_0^b5x^2dxdydz$

I tried solving this problem: Evaluate $\iint_Sx^3dydz+x^2ydzdx+x^2zdxdy$ Where S is the surface bounded by $z=0, z=b, x^2+y^2=a^2$ Using Green Theorem, ...
-1
votes
1answer
21 views

Does these inequalities hold in General for probability distribution? [on hold]

Let $Q(y)$ be a probability density of $y \in [-1,1]$. Then for $t> 0$, the inequalities are $\displaystyle \int_{0 \leq y <t} y^2 Q(y) \, dy \leq t^2 \int_{0 \leq y <t} Q(y) \, dy $. ...
1
vote
0answers
32 views

Reasons for different answers when finding area using Simpsons rule and numerical integration?

I have a function $\sqrt{x^4(x+4)}$ to be integrated from 0 up to -4. Using Simpson's will give me 19.02 but using normal numerical methods giving me -19.5 ! What's the reason behind this difference ...
0
votes
1answer
30 views

an integral problem

$$ \int_{-\infty}^{\infty} [c_1 + c_2 (x-c_3)^2 + (x - c_4)]^{-c_5} \, dx $$ with $c_1, c_2, c_3, c_4, c_5$ known real constant. Can you help me to solve this integral?
1
vote
0answers
29 views

Integral of an expression involving sine and cosine powers

For integers $a,n\in \mathbb N$, consider the following integral $$ I_n(a) = \frac{(-i)^x}{\pi}\int_0^\pi e^{i\theta(n-2a)} \sin^x \theta \cos^{n-x} \theta\; \mathrm d\theta\;. $$ How would one go ...
0
votes
2answers
18 views

Definite integral question

Let $ f(x)$ be a quadratic equation with $f'(3)=3$. If $I=\int_{0}^{\frac{\pi}{3}}t \times \tan(t)dt $ and the value of integral$\int_{3-\pi}^{{3+\pi}}f(x) \times \tan(\frac{x-3}{3})dx $ is equal to ...
0
votes
2answers
44 views

How is $ \frac{\sqrt{a}}{a+1} (0^{a+1}+1^{a+1}) $ equal to $ \frac{\sqrt{a}}{a+1} (-1)^a $

I am trying to integrate this equation $$ y = \int_{-1}^0 \sqrt{a} x^{a} $$ $$ y = \sqrt{a} \int_{-1}^0 x^{a} $$ $$ y = \frac{\sqrt{a}}{a+1} \int_{-1}^0 x^{a+1} $$ $$ y = \frac{\sqrt{a}}{a+1} ...
-2
votes
1answer
25 views

Find value of define integral with [on hold]

Hi i need help for this problem, i very appreciate your sugerences. $$F(x)\text{=}\int ^{g(x)}_{0}\frac{dt}{\sqrt{1+t^{2}} } $$ And $$g(x)\text{=}\int ^{\cos x}_{0}[1+\sin t^{2}]dt$$ For $F'(π/2)$.
5
votes
2answers
353 views

demostration of interger part integration.

I need help for solving this demostration, I appreciate your suggestions very much. $$\begin{array}{rclr} \int ^{n}_{0}[x] dx= \frac{n(n-1)}{2} \end{array}$$ Pd. If you have any suggestion of a ...
4
votes
4answers
90 views

Am I getting the right answer for the integral $I_n= \int_0^1 \frac{x^n}{\sqrt {x^3+1}}\, dx$?

Let $I_n= \int_0^1 \dfrac{x^n}{\sqrt {x^3+1}}\, dx$. Show that $(2n-1)I_n+2(n-2)I_{n-3}=2 \sqrt 2$ for all $n \ge 3$. Then compute $I_8$. I get an answer for $I_8={{2 \sqrt 2} \over 135}(25-16 ...
1
vote
1answer
39 views

Double integration

Evaluate $$ \iint \limits_R(2xy+9) \;\mathrm{d}A $$ where $R$ is the region bounded by $y=x^2$ and $y=x+2$. I have drawn my picture and have come up with my regions from $\sqrt{y}$ to ...
1
vote
1answer
41 views

Find the average value of $f(x,y)=2e^{y}\sqrt{x+e^{y}}$ on the rectangle with vertices $(0,0), (4,0), (4,1),$ and $(0,1).$

Find the average value of $f(x,y)=2e^{y}\sqrt{x+e^{y}}$ on the rectangle with vertices $(0,0), (4,0), (4,1),$ and $(0,1).$ I keep getting a really messy integration.
1
vote
0answers
15 views

Infinite encirclement of branch cut

Consider the integral $$I=\int _\Gamma\frac{1}{4+i(\log z)^2}dz$$ Where $\Gamma$ encircles the unit circle infinitely many times. Would it then make sense to use a parameter n: encirclement count, ...
0
votes
1answer
41 views

Integration with respect to dx, dy and dz (More than one variable)

Sorry if my title was vague but i was not entirely sure what its called. Anyways i was solving some work and energy problems and encountered this integration: $$\int_{2,1,4}^{2,-3,3} 2x\sin^2y ...
0
votes
1answer
68 views

How to integrate $\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$?

How to integrate: $$\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$$ I don't really have a clue? Do I need to simplify it first somehow?
1
vote
0answers
21 views

On utilizing the Leibniz rule of integration on a non compact interval.

I am following some slides that you can find here. At slide $\approx$ 24 a problem arises, to find $$\DeclareMathOperator*{\argmin}{\arg\!\min} \argmin_{\hat{y} } -\int_{-\infty}^{\hat{y}} (y ...
3
votes
2answers
122 views

Confused about calculating the area under the curve

What is the area under the curve of the following function? $f(x) = x² + 2x -3$ $x=-4$ $x=2$ Please, I'd like to see an image. Here is the graphic: https://www.desmos.com/calculator/oe0ja17spg
5
votes
5answers
82 views

$\int\dfrac{dx}{x^2(x^4+1)^{3/4}}$

Evaluate $$\large{\int\dfrac{dx}{x^2(x^4+1)^{3/4}}}$$ I thought of rewriting this as $$\large{\int\dfrac{dx}{x^5(1+\frac{1}{x^4})^{3/4}}}$$ and substituting ...
0
votes
0answers
21 views

Are the polynomials that are orthogonal in the continuous case, still continuous in the discrete case?

One of my friends asked me this question. "Are the polynomials that are orthogonal in the continuous case, still continuous in the discrete case?" It is curious how even the most trivial questions ...
1
vote
0answers
28 views

Cubic Approximation to $e^x$ using Chebyshev Polynomial

Was trying to solve this: $C_r=\frac{2}{\pi}\int_{-1}^1\frac{e^xT_r(x)}{\sqrt{1-x^2}}dx$ where $r=0,1,2,3$ $T_r(x) =cosr[{cos}^{-1}x]$ While solving, I equated $x=cos\theta$ Therefore ...
2
votes
0answers
55 views

How to integrate the following sum?

I'm currently trying to show: $$ \int_0^1{\int_0^y{\sum_{n=0}^{\infty}\left(\frac{1}{10^{n+1}x(1-x)}\left(9+\frac{1}{1-x^{10^n}}-\frac{10}{1-x^{10^{n+1}}}\right)\right)dx}dy}=\frac{10}{99}\log(10) $$ ...
0
votes
1answer
91 views

Calculating in closed form $\int_0^1 \log(x)\left(\frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}}\right)^2 \,dx$

What real tools excepting the ones provided here Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $ would you like to recommend? I'm not against them, they might ...
4
votes
7answers
127 views

Evaluating numerically $\int_0^{\infty}e^{-t^2 /100} \sin \pi t $

What is an appropriate method to approximate $$I=\int_0^\infty e^{-t^2 /100} \sin \pi t \ dt?$$ This is for a Physics problem, but in fact I need this in general, as my professor and book taught us ...
10
votes
2answers
139 views

Integral $\int_0^1\frac{\log(x)\log(1+x)}{\sqrt{1-x}}\,dx$

I'm trying to evaluate this definite integral: $$\int_0^1\frac{\log(x) \log(1+x)}{\sqrt{1-x}} dx$$ It's clear that the result can be expressed in terms of derivatives of a hypergeometric function with ...
1
vote
4answers
136 views

$U_n= \int_{0}^{1}\frac{1}{1+x^{n}}dx$

$U_n= \int_{0}^{1}\frac{1}{1+x^{n}}dx$ where Find $\lim_{n\to \infty} U_n$ can i enter the limit inside ? $W_n= \int_{0}^{1}\frac{x^n}{1+x^{n}}dx$ and i established this relation by parts: $W_n= ...
2
votes
1answer
51 views

integrals of exponential functions over the real axis

How to evaluate the integral $$ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})dx? $$ I intend to change the variable $x$ to $\tan t$ but failed... How to solve it?
1
vote
0answers
103 views

A double integral consisted of hypergeometric functions [on hold]

Calculate in closed form $$\small\int _0^1\int _0^{\infty }\left(-\frac{9 \sqrt{\frac{3}{\pi }} \Gamma \left(\frac{4}{3}\right) \Gamma \left(\frac{5}{3}\right) \, ...
8
votes
3answers
105 views

Logarithmic Integral II

While reviewing an old calculus book the following integral was assigned: \begin{align} \int_{0}^{1} \left( x^{a-1} - x^{n-a-1} \right) \, \frac{\ln^{2}x \, dx}{1-x^{n}} = \frac{2 \, \pi^{3} \, ...
3
votes
1answer
42 views

Solving an Iterated Integral

Given the iterated integral: $$\int^{\sqrt{2}}_{-\sqrt{2}}\int^{\sqrt{2-x^2}}_{-\sqrt{2-x^2}}\int^{\sqrt{4-x^2-y^2}}_{\sqrt{x^2+y^2}}{\left(x^2+y^2+z^2\right)^{3/2}}dzdydx$$ Now, my question is, ...
1
vote
1answer
79 views

Calculating in closed form an integral in Airy function

Can we hope for a nice closed form for the integral below? $$\int_0^1 \frac{\displaystyle \text{Ai}\left(-\frac{t}{2^{2/3} \sqrt[3]{3-3 t}}\right)^2+\text{Bi}\left(-\frac{t}{2^{2/3} \sqrt[3]{3-3 ...
2
votes
0answers
41 views

how to simplify the integral: $\int_{0}^{\infty} \frac{\sin x t \sin y t}{t-z} dt$ , $z= a+ib, a>0,b>0, x >0 ,y >0$

$$\int_{0}^{\infty} \frac{\sin x t \sin y t}{t-z} dt$$ where $z= a+ib, a>0,b>0, x >0 ,y >0$. It is known that: $ \int_{0}^{\infty} \frac{\sin{xt} \sin{yt}}{t} dt = ...
2
votes
3answers
45 views

Definite Integration with Trigonometric Substitution

I'm working on a question that involves using trigonometric substitution on a definite integral that will later use u substitution but I am not sure how to go ahead with this. ...
-2
votes
1answer
48 views

Transforming the probability distribution that have unknown form [closed]

I have the following expression, which is difficult to compute as the explicit form of the probability distribution is unknown. $\int_{0 \leq y < t} y^2 Q(y)dy$. The density $Q(y)$ is for $y ...
0
votes
1answer
28 views

Question about relation between integral and summation in this case

I have $N(t)=N_{\alpha_{n}}(t)=\#\lbrace n:\alpha_{n}\leq t\rbrace$. Let $\{\alpha_{n}\}$ be a positive sequence, tending to infinity. Let $ \varphi (t) $ be a differentiable, positive, and ...
8
votes
3answers
86 views

Is there a direct method for evaluating this integral: $\int_{0}^{2\pi}\ln^2(2\sin(\frac{x}{2}))dx$?

I stumbled upon this integral while attempting to evaluate $\sum_{n=1}^{\infty}\frac{\cos(n\theta)}{n}$. I started with the series $-\ln(1-z)=\sum_{n=1}^{\infty}\frac{z^n}{n}$, replaced z with ...
3
votes
1answer
45 views

Is this function in $L^1_{loc}(\mathbb R^3)$

It seems such a trivial question, but for whatever reason I don't understand. Let $u: \mathbb R^3 \to \mathbb R $ be $$u(x) = \frac 1{4\pi |x|}$$ The book says that $u \in L^1_{loc}(\mathbb R^3)$ ...
3
votes
2answers
85 views

Integrate area of the shadow?

Today I found an interesting article here. It computes (approximately) area of the shadow. I was wondering what is exact value of the area. My first thought was to use integrals but it doesn't seem ...
27
votes
1answer
1k views

How is the derivative geometrically inverse of integral?

I know that derivative is the slope of the tangent line, and that integral is the area under the curve. My question is that how these two distinct concepts are geometrically related? What is the ...
2
votes
1answer
87 views

Improper integral of $\sqrt{x^4 + 1} - x^2$ [duplicate]

I'm having a little trouble with this integral: $\int^\infty_0 (\sqrt{x^4+1} - x^2)\,dx$. Using the likes of Maple, I can easily find that it takes the form $-\frac{2}{3}\sqrt{2}(1+i)K(i) - ...
2
votes
1answer
111 views

Compute this integral

$$ \displaystyle \int_{0}^{\infty}{\frac{{log}^{2}(1-{e}^{-x}){x}^{5}}{{e}^{x}-1} dx} $$ What I have done - $ \displaystyle I(k) = \int_{0}^{\infty}{\frac{{x}^{5}}{{e}^{x}{(1-{e}^{-x})}^{k}}}$ ...