Questions about the evaluation of specific definite integrals.

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1answer
15 views

Area of the region inside $r = 1 - \cos(\theta)$ and also inside $r = \cos(\theta)$

Pretty simple polar integration question that I've been having trouble with... The question says it all. I identified the limits of integration by setting $1 - \cos(\theta) = \cos(\theta)$ so that ...
0
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0answers
11 views

How to compute $\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$

For a project I want to get a closed form solution of $$\int_{-1}^1 x^p (1-x^2)^{\frac{d-3}{2}} P_n^d(x) dx$$ Here $p \in \mathbb{N},\; d\ge3, \; d\in\mathbb{N}$ and $P_n^d$ is the associated ...
6
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1answer
50 views

How to calculate the area between $y=e^{-x}$, $y=x$ and $x=0$

My problem is that little point that comes from the equation $$e^{-x} = x$$ I can't solve that one. Is there another way without knowing that point or a way to calculate it? Thanks in advance!
4
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2answers
59 views

Easier way to solve $\int_0^1 \frac{dx}{\lfloor{}1-\log_2(x)\rfloor}$

This problem showed up in the MIT integration bee last year: $$\int_0^1 \frac{dx}{\lfloor{}1-\log_2(x)\rfloor}$$ Basically, after doing a lot of tedious work I graphed out part of the function and ...
1
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0answers
27 views

Stokes' theorem to evaluate an integral

Let $C$ be the intersection curve of the parabolic sheet $y=x^2$ with the cylinder $x^2+z^2=4$, oriented clockwise when viewed from the positive $y$-axis. Apply Stoke's Theorem to the integral $$ ...
5
votes
3answers
81 views

Putnam definite integral evaluation $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$

Evaluate $$\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$$ Source : Putnam By the property $\displaystyle \int_0^af(x)\,dx=\int_0^af(a-x)\,dx$: $$=\int_0^{\pi/2}\frac{(\pi/2-x)\sin ...
5
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2answers
70 views

Prove that $\int_0^\infty\frac{x^n}{1+e^{x-t}}\mathrm{d}x = \frac{t^{n+1}}{n+1} + o(t^n)$, when $t \to \infty,\,n\in\Bbb{R}^+$

I hae to prove that $$\int_0^\infty\frac{x^n}{1+e^{x-t}}\mathrm{d}x = \frac{t^{n+1}}{n+1} + o(t^n), \quad\text{ when } t \to \infty,\,n\in\Bbb{R}^+$$ where $o(\cdot)$ is the Little-o notation. What ...
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2answers
28 views

Finding the mass of a cone using triple integral

I have a density $\rho(x,y,z) = 3-z$ and have converted my given information to form a triple integral equation for finding the volume of my cone in cylindrical coordinates and have found the volume ...
0
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1answer
34 views

Volume bounded by two solids

Can somebody help me get started in the right direction for this question involving volume? The question is "Find the volume of the solid region inside the hemisphere $x^2 + y^2 + z^2 =6, z<0$ but ...
1
vote
1answer
43 views

Evaluate $\int_0^1\int_x^1 e^{x/y} dy\,dx$

I need some help to solve the following: $$\int_0^1\int_x^1 e^{x/y} dy\,dx$$ I guess it is related with change of variable, but I can't figure out which one. Thanks in advance. Regards.
2
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1answer
43 views

Double integral with two parameters $\int_{x_1=1}^{x_2=2}\int_{y_1=0}^{y_2=x}\arctan\left(\frac{y}{x}\right)\,dx\,dy$

Given the following integral: $$\int_{y_1=0}^{y_2=x}\,dy\int_{x_1=1}^{x_2=2}\arctan\left(\frac{y}{x}\right)\,dx$$ I thought of using $u$-substitution: $$\begin{align} u &= \frac{y}{x} \\ w ...
1
vote
1answer
20 views

Volume of a solid formed as vertical limit goes to infinity

Here's the question: The way I have it set up currently is as follows: $V = \pi \lim_{a \to \infty} \int_1^a (a-1)^2 - (\frac{1}{\sqrt{x^5}} - 1)^2$ But how do I go from here? And is the working ...
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2answers
64 views

Intuitive but hard question about an integral?

Let $f \colon [0,1]\rightarrow \mathbb{R}$ be a function with continuous derivative such that $f(1)=1$. Evaluate $$\lim_{y\rightarrow \infty}\int_0^1yx^yf(x)dx$$
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0answers
11 views

Any way to simplify integral of Confluent Hypergeometric Function of the First Kind?

The integral is this: $$\int_{-\log n}^{0}e^{t(1-s)} \cdot z \cdot {}_1F_1(1-z, 2, t) dt $$ Is there a way to write this in terms of special functions that eliminates the integral and doesn't use ...
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0answers
13 views

Is it always possible to use Chasles to decompose an integral ?

Well, I'm in "classe préparatoire" and I always learn that if f is a continuous fonction integrable on [a,b] and if c is in [a,b] then with Chasles relation we have : $$ \int_a^b f(x) dx = \int_a^c ...
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0answers
36 views

How do we calculate naturally the following integral :

Do you know a natural method to calculate the following integrals: $$ I = \int_{\mathbb{P}^{1} (\mathbb{R})} \dfrac{1}{x} dx\quad\text{and}\quad J = \int_{\mathbb{P}^{1} (\mathbb{C})} \dfrac{1}{z} dz. ...
0
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1answer
36 views

Is it possible to find the closed-form expression for $\int_{\alpha}^{\infty} \frac{e^{-At}}{\left(1+ Bt\right)t^m}dt$?

Is it possible find the closed-form expression or represent it in other function for this integral: \begin{align} I= \int \limits_{\alpha}^{\infty} \frac{e^{-At}}{\left(1+ Bt\right)t^m}dt ...
3
votes
1answer
40 views

Handling integrals of trig functions

I'm not sure how to handle the following class of integrals: $I=\int_0^{2\pi}f(\cos(\theta))d\theta$ If I make the change of variables $x=\cos(\theta)$ the new limits of the integral are the same, ...
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2answers
66 views

Solving $ \int_{0}^{2\pi} e^{-x} \lvert \sin x\rvert,dx $

Since it involves an absolute value, I assume I need to split it into two cases? For $ 0 \le x \le \pi $ $$ \int_{0}^{\pi} e^{-x} \sin x\,dx $$ and for $ \pi \le x \le 2\pi $ $$ \int_{\pi}^{2\pi} ...
2
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1answer
35 views

How to evaluate a double integral with two Dirac functions?

Here I have a problem, is the solution the same if I integrate every one? part by part? $$\int_0^Te^{-(s+\mu\lambda^2 ) t} \int_0^l\left[\delta(x-R)\delta(t-tj)\varphi(x) \, dx\, dt\right]$$ I've ...
2
votes
2answers
47 views

Taylor Expansion of $x\sqrt{x}$ at x=9

How can I go about solving the Taylor expansion of $x\sqrt{x}$ at x=9? I solved the derivative down to the 5th derivative and then tried subbing in the 9 value for a using this equation ...
15
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5answers
227 views

Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$

Today I discussed the following integral in the chat room $$\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$$ where $0\leq a, b\leq \pi$ and ...
1
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1answer
33 views

Riemann integrability of a square of a continuous function

Let, $f(x)$ be continuous in $[0,1]$ such that, $\int_{0}^{1}x^{n}f(x)dx=0$ for $n=0,1,2,3,...$. Then prove that, $\int_{0}^{1}f^{2}(x)dx=0$. First we apply $1^{st}$ M.V.T. of integral calculus & ...
4
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1answer
59 views

Reduction formula of primitive $\big(1-\sin^3{x}\big)^n\cos{x}$

I am trying to obtain a reduction formula for $$\int_0^{\pi/2}\big(1-\sin^3{x}\big)^n\cos{x}\;\mathrm dx $$ where $n \in \mathbb{N}$. My attempt is as follows $$\text{let } v = \sin{x}\; \implies ...
5
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0answers
64 views

Find a function that maximizes $\int_{0}^{1}f(x)\,\rm dx$ with given constraints

Find a function $f(x)$ that maximizes the following integral $$\max\int_{0}^{1}f(x)\,\rm dx\quad \text{s.t.}\quad \frac{d}{dx}ln(f(x))<0$$ $f(x)$ also continues, $f:[0,1]\rightarrow R$ and we ...
1
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1answer
37 views

Growth of a sequence

Let $$a_n=\int_{\frac{\pi}{2}+n\pi}^{ \frac{3\pi}{2}+3n\pi}\frac{\cos{t}} {t} dt$$ How to show that $\left(a_{2n}\right)_{n\geq 0}$ is increasing (strictly) and $\left(a_{2n+1}\right)_{n\geq 0}$ is ...
4
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4answers
150 views

How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$

I found this nice result. Prove that $$\int_{0}^{\infty}\sin{x}\arctan\left({\frac{1}{x}}\right)\,\mathrm dx=\frac{\pi }{2} \left(\frac{e-1}e\right)$$ I tried some methods but I can't ...
4
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1answer
47 views

Calculate $\int_{|z|=1}\frac{dz}{\sin z}$

I have to evaluate $\int_{|z|=1}\frac{dz}{\sin z}$. Any tips?
1
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1answer
18 views

Object moving on x axis integration

So, I think I know how to set up this problem, but then I get stuck at the last part: An object moves along the x - axis such that its velocity at time $t$ is $v(t) = cos(2t) $. Suppose the object ...
10
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4answers
123 views

Evaluating $\int_{0}^{\pi/2}\frac{x\sin x\cos x\;dx}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}$

How to evaluate the following integral $$\int_{0}^{\pi/2}\frac{x\sin x\cos x}{(a^{2}\cos^{2}x+b^{2}\sin^{2}x)^{2}}dx$$ For integrating I took $\cos^{2}x$ outside and applied integration by parts. ...
1
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2answers
29 views

area under a curve and units

If we introduce a unit of length like meter for $x$ and integrate the function $f(x)=x^2$ from $0$ to $2m$ we get $\dfrac{8}{3} m^3$. How can this be interpreted geometrically? My initial thought was ...
9
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3answers
122 views

How to find $\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$

Evaluate $$\int_{-1}^1 \frac{\cos x}{a^x+1}\mathrm dx$$ where $a$ is a real parameter $a\geq1$. I can easily find the definite integral for $a=1$. It is $\sin(1)$. In wolframalpha.com when I put ...
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1answer
33 views

Evaluate integral by interpreting it in terms of areas

I tried (a) and I got 5, but I am suppose to get a 4. I really need a good explanation to understand how to approach these problems. I tried searching in youtube and stuff, but it was not helpful. ...
5
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1answer
60 views

An advanced integral $\int_0^1 \frac{(2 e)^{-1/y} \left(2 e^{1/y}-e 2^{1/y}\right)}{1-y} \ dy$

I'd like to ask you how you would like to approach the integral below $$\int_0^1 \frac{(2 e)^{-1/y} \left(2 e^{1/y}-e 2^{1/y}\right)}{1-y} \ dy$$ and then recommend me some tools you'd employ. It's ...
0
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1answer
22 views

Elementary question: Integral of area function

I am sorry for this elementary question. I have searched a bit but haven't found what I am looking for precisely. I am trying to determine how to the volume of liquid in an irregularly shaped ...
2
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2answers
70 views

A reduction formula for $\int_0^1 x^n/\sqrt{9 - x^2}\,\mathrm dx$

Let $$I_n = \int_0^1 \frac{x^n}{\sqrt{9 - x^2}}\,\mathrm dx$$ Using integration, show that $$nI_n = 9(n - 1)nI_{n - 2} - 2\sqrt2$$ I've found that $\displaystyle I_0 = ...
2
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0answers
26 views

Determining solid region from bounds of triple integral

If you have an integral such as: $$\int_0^1\int_0^{2-x^2}\int_0^{2-x}f(x,y,z)dydzdx$$ How can you determine the equation for the solid region represented by the bounds of this triple integral? Does ...
3
votes
2answers
72 views

How to evaluate $\int_0^1 \frac{2-t}{t^2-t+1} dt$?

How to evaluate $$\int_0^1 \frac{2-t}{t^2-t+1} dt\;?$$ I tried doing it using $s=-t+1$ but it wasn't useful. We've learned in class that having a polynomial in the denominator is considered to be ...
0
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0answers
47 views

Definite integral involving Error function

Let us write $$\mathrm{erf}(x)=\frac{2}{\sqrt {\pi}}\int_0^x e^{-t^2}dt $$ for the usual Gauss error function. Given natural numbers $m,n,k$ I am interested in computing the integral ...
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2answers
32 views

Hint for the integral

I was trying to compute the following integral but got stuck at the starting point. Can anyone provide a valuable hint for the evaluation of this integral $\int_{0}^{\infty}x^{9}e^{-x^{2}} dx$ ?
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2answers
75 views

Is it possible to evaluate $\int_0^1 \sin(\frac{1}{t})\,dt\,$?

I was wandering if it possible to evaluate the value of the following improper integral: $$ \int_0^1 \sin\left(\frac{1}{t}\right)\,dt $$ It is convergent since $\displaystyle\int_0^1 ...
0
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1answer
19 views

Solving a line integral - splitting up in int multiple “segments”

I have the following formula: $$f = \oint\frac{ds}{C}$$ This integral happens over a (closed) circle with radius $r$, so normally the solution would be: $$\oint\frac{ds}{C} = \frac{2\pi r}{C}$$ ...
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0answers
35 views

Water force on the wall of a dam using double integrals

The wall is given by the function $kx^2, g=9.8 m/s^2, p=1000kg/m^3, F/A=gp(H-y)$ and is the force of water exerted on the dam. y is the height in meters above the base. Total force is ...
0
votes
2answers
47 views

Evaluate $\int_3^9 \frac{1}{x \log x} \,\mathrm dx$

$$\int_3^9 \frac{1}{x \log x} \,\mathrm dx$$ I tried : $$u=\log x \implies x \ln 10 du = dx$$ $$\ln 10 \int_{\log 3}^{2\log 3} \frac{1}{u} du = \ln 10 \left[\ln u \right]_{\log 3}^{2\log 3}=\ln 10 ...
0
votes
1answer
62 views

Evaluate $\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$

$$\int_0^4 \frac{\ln x}{\sqrt{4x-x^2}} \,\mathrm dx$$ How do I evaluate this Integral?
3
votes
2answers
90 views

Evaluate $\int_0^{\pi}\frac{x\sin x}{1+\cos^2x}\,\mathrm dx$

I can't seem to get any kind of result which I find useful when I use the fact that $$\int_0^af(x)\,\mathrm dx=\int_0^af(a-x)\,\mathrm dx$$ After using trigonometry, I end up getting: ...
0
votes
1answer
25 views

Work required to pump water out of tank in the shape of a paraboloid of revolution

This is the problem I have been assigned: A water tank has the shape of a paraboloid of revolution: its shape is obtained by rotating the parabola $y=x^2/4$, for $0\le x\le 4$, around the ...
2
votes
1answer
53 views

Question about length of curve?

The question: Find length of curve defined by $\displaystyle y=2\ln\left[\left(\frac{x}{2}\right)^2-1\right] $ from $x=4$ to $x=6$ Here is the work I have done, but I seem to keep getting it ...
0
votes
1answer
54 views

Find $\int_2^{2.2}f(x)\,\mathrm dx$ given $f(x)=x^4-3x^3+9x^2+22x+6$.

$f(x)=x^4-3x^3+9x^2+22x+6$. Find $\int_2^{2.2}f(x)dx$ by finding $f(x-2)$ This is in a non-calculator paper which is why $f(x-2)$ is meant to be obtained (it's supposed to made the maths possible to ...
9
votes
1answer
103 views

Is the integral always the area under the curve?

It might be a stupid question but if I were to ask to compute the definite integral $$\int_{\frac{- \pi}{2}}^{\frac{\pi}{2}} \sin(x) \ dx$$ then on plugging the values then I would get "$0$" as the ...