Questions about the evaluation of specific definite integrals.

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6
votes
3answers
147 views

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...
0
votes
1answer
34 views

Findng the area under the curve $y=3-3\cos(t),x=3t-3\sin(t)$

I need to find the area under the curve $\color{blue}{y=3-3\cos(t),x=3t-3\sin(t)}$ and between $\color{blue}{x=2\pi,x=0\text{, above axis}}$ using $\color{blue}{\text{Green's theorem}}$. My attempt ...
1
vote
1answer
33 views

What is the solution to this integral?

In some calculation, I encounter an integral of the form \begin{equation} \int_{-\infty}^\infty \text dz\ \frac{1}{z-i\varepsilon}e^{- a z^2+i b z}, \end{equation} where $a>0$ and $b$ are some ...
1
vote
2answers
24 views

Finding the flux of $\iint \vec F\hat n\;ds$

I need to find the flux $\displaystyle\iint \vec F\hat n\;ds$ of the vector feild $\vec F=4x \hat i-2y^2\hat j+z^2 \hat k$ throughe the surface $S=\{(x,y,z):x^2+y^2=4,z=0,z=3\}$ My attempt: (I'm ...
2
votes
1answer
36 views

How to prove that $f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt},$ for all invertible functions.

A while ago, I found that: $$f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt}.$$ I managed to prove it for a few functions, and I believe that it may be the case for all ...
1
vote
0answers
26 views

Help solving integration: $I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(a/\sqrt{b+c\mathrm{e}^{\frac{x-\mu}{\sigma}}}\right)dx$

My work has arrived at needing to solve the integral below for $a,b,c,\sigma>0$ $$I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(\frac{a}{\sqrt{b+c\mathrm{e}^{(x-\mu)/\sigma}}}\right)dx$$ I ...
0
votes
2answers
108 views

How to find $\int x^2e^{x^2}dx$?

How to find $\int x^2e^{x^2}dx$? I tried integration by parts following ILATE rule but it's not working.Please help!! What should I take as first function ? If it's not integrable can you atleast ...
3
votes
3answers
78 views

Evaluate $\iint_{R}(x^2+y^2)dxdy$

$$\iint_{R}(x^2+y^2)dxdy$$ $$0\leq r\leq 2 \,\, ,\frac{\pi}{4}\leq \theta\leq\frac{3\pi}{4}$$ My attempt : Jacobian=r $$=\iint_{R}(x^2+y^2)dxdy$$ $$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$ ...
2
votes
4answers
150 views

How should I go about solving this definite integral?

The integral is: $$\int_{-1}^1\sqrt{4-x^2}dx$$ I'm having difficulty figuring out how to go about this. I attempted to use u-substitution, both by substituting $u$ for $\sqrt{4-x^2}$ entirely, and ...
3
votes
3answers
156 views

Definite integral with limits from zero to infinity

Let $ I=\int\limits_{0}^{\infty}e^{-(x^2+\frac{1}{x^2})}dx$ and $J=\int\limits_{0}^{\infty}x^2e^{-(x^2+\frac{1}{x^2})}dx$. If $J=\dfrac{pI}{q}$, then find the value of $p+q$ where $p$ and $q$ are ...
0
votes
1answer
25 views

Associate Legendre polynomials of first and second kind; the integral relastionship

The Legendre functions of first $P_n(x)$ and second $Q_n(x)$ kind are related by the definite integral $$ Q_n(x) = {1\over 2} \int_{1}^{-1}{P_n(x) \over u-x}\, du. $$ The associated Legendre ...
4
votes
3answers
196 views

Not the toughest integral, not the easiest one

Perhaps it's not amongst the toughest integrals, but it's interesting to try to find an elegant approach for the integral $$I_1=\int_0^1 \frac{\log (x)}{\sqrt{x (x+1)}} \, dx$$ $$=4 ...
1
vote
4answers
78 views

Evaluate $\iint dy\,dx;\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4};0\leq r\leq2$

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4}\;\;\;\;,0\leq r\leq2$ $\color{blue}{\text{without using polar coordinates}}$. My attempt: ...
0
votes
1answer
23 views

Finding the equivalent definite integrals

$\int x^{m-1}(1-x)^{n-1} dx $ (x=0 to x=1 ) I came across this question.Option A I could prove correct .But is any of the other three options correct? What should be the approach ?
2
votes
0answers
68 views

Can $\int_{0}^{1}\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx$ be expressed in a simple form?

I was browsing the book Irresistible Integrals and found this gem, at page 97, $$ \int_{0}^{1}x^{n}\ln^{k}(x)dx=\frac{(-1)^{k}k!}{(n+1)^{k+1}} $$ that resembles a previous question of mine here. So, ...
0
votes
1answer
32 views

Evaluate $\int_{-2}^{2}\int_{y^2-3}^{5-y^2}dxdy$ [duplicate]

In the black I evaluated the integral and I got 64/3, now I need to evaluate the same integral with $\color{red}{dydx}$ .in the $\color{blue}{\text{blue}}$ color is my attempt, I don't think that my ...
3
votes
4answers
96 views

Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\frac{3}{8}{(\frac{\pi}{\alpha^5})}^\frac{1}{2}$ [duplicate]

In my physics course this standard formula is used a lot without proof so it would be interesting to see a neat proof for it. From a previous thread by me I know the proof for $\int ...
0
votes
1answer
52 views

Definite integral: $\int^\pi_0 e^{2a \cos x} \left( \frac{\sin^2 x}{1- \cos x} \right) dx$

The goal is to solve this: $$ \int^\pi_0 e^{2a \cos x} \left( \frac{\sin^2 x}{1- \cos x} \right) dx $$ with $a>0$. Really not sure how to attack this one. The integrand seems to be capable of ...
2
votes
5answers
60 views

Show that $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2}$ + Constant

I tried to do this integration by parts and got $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2} +\alpha\int x^3\mathrm{e}^{-\alpha x^2}\mathrm dx$ + constant. ...
3
votes
1answer
26 views

How to prove define integrate from f(sin x)

i need help for prove this problem , i dont have idea for this prove, i very appreciate your sugerences. $$ \int ^{\pi }_{0}xf(\sin x)\,dx = \int ^{\pi }_{0}\frac{\pi }{2} f(\sin x)\,dx $$
0
votes
1answer
35 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 = ...
5
votes
1answer
59 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
17 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 ...
4
votes
4answers
126 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
3
votes
0answers
69 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} ...
9
votes
0answers
87 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 ...
1
vote
0answers
94 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
51 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
62 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
29 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
28 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
31 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
1answer
49 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?
2
votes
0answers
35 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
21 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
45 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
27 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
359 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
93 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
43 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
43 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
42 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
69 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
83 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
22 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
42 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 ...