Questions about the evaluation of specific definite integrals.

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Integral of product spherical harmonics

I'm trying to calculate this integral: $$\int_0 ^\pi\int_0^{2\pi} [\sin\theta(\cos\theta \sin \phi + \sin\theta \cos(2\phi))]^2 \sin\theta d\phi d\theta $$ We could compute this integral ...
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2answers
30 views

Recursive formula for definite integral

The integral is: $$I_n = \int_0^{\pi/4} \tan^{2n}x\,dx$$ I'm supposed to find the recursive formula.
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1answer
33 views

The volume of a cone whose length of its side is R

How can i compute the volume of a cone whose length of its side is $R$ and the vertex of the cone forms an angle $2 \theta$ . The top cone is a cap of a sphere of radius $R$. Some help please.
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62 views

Delicate Integral $I=\int_0^\infty \frac{\log^2 x \cos ax}{x^n-1}dx$

Hi I am trying to calculate $$ I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}dx,\quad \Re(n)>1, \, a\in \mathbb{R}. $$ Note if we set a=0 we get a similar integral given by $$ ...
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0answers
50 views

Fancy Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
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2answers
181 views

Definite integral problem

I was solving a definite integral problem which was reduced to : $$\int^{1}_{0} \frac{\ln(1+t)}{t} dt$$ I couldn't solve it and when I saw the solution, the answer was simply given as ...
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3answers
64 views

What is the average value of $y=x^2\sqrt{x^3+1}$ on the interval $(0,2)$

I know the average value formula is $$\frac{1}{b-a}\int_a^b f(x)\;dx$$ I have no problem plugging in 0 and 2 for a and b respectively. I think i'm struggling actually taking the integral.
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0answers
32 views

Area interpretation of integrals

When integrating under part of a circle, as in $$A=\int_0^a {\sqrt{r^2-x^2}\,\mathrm{d}x}$$ I noted that the simple geometric solution would be to add the areas of the sector and triangle formed by ...
2
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1answer
28 views

How to evaluate the definite integral $\int_0^x t^{n-1}e^{-(a+bt)}dt$

How to evaluate the following definite integral $\int_0^x t^{n-1}e^{-(a+bt)}dt$, where $n\in\mathbb{N}$ and $a,b>0$.
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3answers
96 views

Why is $\displaystyle\int^{\infty}_{0}{(1-\cos x)\over{x^{2}}}dx = \frac\pi{2}$?

I have been having trouble understanding Fourier series and analysis in one of my classes. This is one problem from the text and we have to show that this is true. I have done other problems related ...
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2answers
31 views

Calculus power series

Hi could anyone help me to solve this. express the function $\int_x ^0 (\sin(t^2)\cdot \cos(t^2))$ as a power series. Because there is two trigo identies I do not know how to combine them to form a ...
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2answers
42 views

Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$ \int_{\gamma} {dz \over (z-3)(z)} $$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...
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0answers
37 views

Line integral - should I parametrize the square?

I have the following $1-\text{form}$ defined: $$\omega = \displaystyle\frac{2xy}{(1-x^2)^2+y^2}\mathrm{dx}+\displaystyle\frac{1-x^2}{(1-x^2)^2+y^2}\mathrm{dy}$$ I'd like to find ...
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1answer
239 views

Cool Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
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1answer
23 views

How to solve this definite integral from fourier series?

I am stuck with this integral: $$\int\limits_0^{2L} \sin \left( \frac{m \pi x}{L} \right) \sin \left( \frac{n \pi x}{L} \right)\, dx.$$ How to solve this integral? In general, can you refer me to ...
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change of variables in Riemann -stieltjes integral

I really need some help on the following problem. Suppose α is a continuous and strictly increasing function on [0, 2] such that α(0) =0, α(1) = 1/3, α(2) = 1. Calculate the following ...
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4answers
39 views

verifying log rules via elementary properties of an integral

my attempt at this question so far, $$\int_1^x \frac{1}{t}dt +\int_1^y \frac{1}{t}dt= 2\int_1^x \frac{1}{t}dt+ \int_x^y \frac{1}{t}dt$$ But I am not sure hwo to prove it from elementary ...
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16 views

Integral Evaluation: Exponential of and Hyperbolic Function

I'm trying to evaluate $$G^{\pm} = \frac{-i}{8\pi^2 X} \partial_X \int_{-\infty}^\infty d\phi e^{i m \left[X \sinh \phi \pm T \cosh \phi \right]}$$ for $T = \pm X$. Where $T, X, m \in \mathbb{R}$ ...
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37 views

Differentiation under integral sign

There is this integral that I used a lot in my research: $$\int_{-\infty}^{\infty}\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}\left(a\left(x-d\right)\right)\,\mathrm{d}x = ...
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2answers
62 views

Integral question with square roots

I can't find the solution for this integral. I don't know why I didn't succeed in solving it. I will be really glad if someone can help me find the answer to the following: ...
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0answers
84 views

integrate a difficult function

I can't solve it. please help! I tried everything. Integration by parts - doesn't work. but maybe I didnt do it right. I tried to substitute , but I'm stuck. $$\int \frac{x}{\cos x}\sin(\tan ...
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1answer
61 views

Evaluate $\int_a^s\frac{(t-s)^n}{t(t-z)^{n+1}}dt.$

Evaluate: $$\int_a^s \frac{(t-s)^n}{t(t-z)^{n+1}}dt,$$ where $n>0$ is an integer, $0<a<s$. I'm not sure what values $z$ should take yet, but any help with the integral would be helpful. I ...
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3answers
65 views

How should I try to evaluate this integral?

Suppose that we are given the following integral: $$\int_0^\pi \sqrt{1+4\sin^2\frac x2 - 4\sin\frac x2}\;dx.$$ (Original screenshot) And the answer is one of these :- $4\sqrt3-4-\frac\pi3$ ...
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0answers
97 views

Beautiful Closed form $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\ln^3 2 ...
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2answers
52 views

Find the length of $y=x^2$ from $x=0$ to $x=1$

$$L=\int_0^1 \sqrt{1+4x^2} dx = \frac{1}{4}\left(\theta+2\sqrt{5}\right)$$ It tells you to use $2x=\sinh\theta$. I've tried the best I could, but I can never get the right side of the equation. Can ...
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2answers
50 views

Problem calculating line integral

I have $\gamma=[0,1]\to\mathbb{R}^3$ defined by $\gamma(t)=(\cos(2\pi t), \sin (2\pi t), t^2-t)\;\forall t\in[0,1]$ and I'm asked to calculate ...
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42 views

A problem about center of mass

Suppose $f(x)$ is positive, increasing and Riemann-integrable on the interval $[a,b]$. Let$$\bar{x}=\frac{\int_{a}^{b}{xf(x)\text{d}x}}{\int_{a}^{b}{f(x)\text{d}x}}.$$Prove ...
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63 views

How to solve this integral ($\int _{\frac{\pi }{6}}^{\frac{\pi }{4}}\sqrt{1-\tan ^2\left(x\right)}dx$)

$$\int _{\pi/6}^{\pi/4}\sqrt{1-\tan ^2\left(x\right)}dx$$ Hey, can you help me to solve this integral please? Thanks.
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41 views

Computing the values of $ W_\alpha(n):=\int_0^\pi x^{\alpha-1}\sin(x)^{2n} $

Define the following integral as $$ W_\alpha(n):=\int_0^\pi x^{\alpha-1}\sin(x)^{n}\,,\quad V_\alpha(n):=\int_0^\pi x^{\alpha-1}\cos(x)^{n} $$ where $n \in\mathbb{N}$. Now in the base case ...
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35 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
3
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2answers
71 views

Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
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1answer
31 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
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1answer
32 views

Re-interpreting double integral as a Type II Region $\mathrm{d}y\,\mathrm{d}x$ vs $\mathrm{d}x\,\mathrm{d}y$

I have the following Double Integral:$\iint_Dx\cos y\space\mathrm{d}A$ where $a$ is bounded by $x=1,y=0,y=x^2$. Interpreting this region as a Type one region, it is easy to conclude $R=\{(x,y)\mid ...
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1answer
38 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
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29 views

Find the power series for a definite integral

I am a bit unsure when integration is used together with summation. Here is my question: Find power series for $\int_0^{1} \frac{\sin x}{x}dx$ in the form $\sum_{k=1}^{\infty} a_kx^k$ Here is what I ...
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1answer
71 views
+50

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
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45 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
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2answers
36 views

Definite integral (mixture of functions)

I have problem with this integral and I generally don't know how to approach it: $$\int_{-2}^2 (x^4+4x+\cos(x))\cdot \arctan\left(\frac{x}{2}\right)dx$$ I know that I probably have to make some ...
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1answer
133 views

How to prove $\int^1_0\int^1_0\frac{\log(x-x^2)-\log(y-y^2)}{(x-x^2)-(y-y^2)}dxdy=7\sum_{i=1}^\infty i^{-3}$?

How do you prove that $$\int^1_0\int^1_0\frac{\log(x-x^2)-\log(y-y^2)}{(x-x^2)-(y-y^2)}dxdy=7\sum_{i=1}^\infty i^{-3}\;\;\;\left(=7 \zeta(3)\right)~?$$ p.s. Mathematica gives a pretty good ...
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+400

$I=-\frac{4}{3}\log^3 2-\frac{\pi^2}{3}\log 2+\frac{5}{2}\zeta(3)$

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
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39 views

For solid volumes, why does the Integral behave as a summation?

When you take a definite integral, you can think about calculating the area under the curve (via Riemann rectangle slices approximation) Now, when you take the volume of a 3D object, you sum the ...
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3answers
71 views

$\int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx.$

Evaluate $\int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx.$ Sidenote:Via mathalpha I know that answer is $-\pi/4$ but do not know how to derive that.
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1answer
47 views

Expansion of Integration

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
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2answers
59 views

Integral $I=\int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx$

Hi I am trying to show $$ \int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx=\frac{\pi}{32\alpha^{3/2}},\quad \Re(\sqrt \alpha)> 0. $$ I am looking for a solution to this NOT using contour integration, but ...
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1answer
34 views

Question regarding trigonometry

I've got this thing on my mind : we know that $cos(x)$ is a periodic function , hence integral from $2(k-1) \pi$ to $2k \pi$ will yield the same value for any $k \geq1$. My question is , why is ...
3
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2answers
36 views

Short argument for asymptotic value of parameter integral

I want to find the main term of the asymptotic expansion for $x\to 0^+$ for $$f(x)=\int_0^{\pi/2} \dfrac{\cos t}{t+x}dt.$$ Now, clearly, the problem is at $t=0$ and the cosine is almost $1$ ...
7
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3answers
113 views

Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$

I was reading an article in which it was stated that, with a change of variable, one could show that: $$\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$$ I tried with $t = 1 + ...
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1answer
36 views

Help me integrate this function using Simpson's rule

I have a question: compute $$\int_0^1 \frac{\sin(x)}{x}\,dx$$ for $n=10$ divisions. I got the value $0.9127$ but I think its a bit too high.
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0answers
16 views

Argument Principle solving the quesion

1/2pi i ∫ c(0,12) f'(z)/f(z) dz where f(z) = (z-5+12i)^4(z-7)^6 / 12(z-8)^6(z-i+6)^7 dz. How do I go about doing this kind of question. I only know its argument principle. I googled it but still cant ...
2
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1answer
53 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...