Questions about the evaluation of specific definite integrals.

learn more… | top users | synonyms (1)

1
vote
1answer
22 views

How to compute $\int^{1}_{-1}f(x)dx$?

I need to compute $\int^{1}_{-1}f(x)dx$, where $$f(x)=\{^{x,\ x \leq 0}_{x^2,\ x \geq 0}$$ I know that $\int^{1}_{-1}xdx=0$ & $\int^{1}_{-1}x^2dx=\frac{2}{3}$ Is it right?
11
votes
0answers
75 views

Evaluate $\int_0^{\pi/2} \frac{\ln\left(e^{2x} + 1\right)}{1 + \sin2x}\mathrm dx$

Here's my Xmas gift to all of you! I just encountered a very tough integral. $$\int_{0}^{\pi/2} \frac{\ln\left(e^{2x} + 1\right)}{1 + \sin2x}\mathrm dx$$ I have tried for a few hours. This task is ...
6
votes
2answers
71 views

Closed form of a Definite Integral [duplicate]

I attempted to integrate the following function from a practice problem in my Calculus textbook: $$\displaystyle \int_{0}^{\frac{\pi}{2}}{\frac{1}{1+\tan^\sqrt{2}(x)}} \ {\rm d}x$$ I failed to find ...
2
votes
1answer
69 views

Evaluating $\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$

How to calculate this integral? $$\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$$ I suppose that it should be parted like this: $$\int_0^{1} \frac{dx}{\sqrt[3]{2x^2-x^3}} + \int_1^{2} ...
9
votes
3answers
128 views

Proving $\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} dx=1-\frac{\gamma}{2}-\ln2$

Nowadays I encounter an integral which is difficult for me to evaluate it. Please help me to evaluate it. Thank you. $$\int_{0}^{\infty}\frac{x}{(x^2+1)(e^{2\pi x}+1)} ...
0
votes
0answers
26 views

How to calculate the volume of a tetrahedron?

Suppose that $$ I=\iiint_{V}f(x,y,z)dxdydz $$ where $f(x,y,z)$ is a continuous function, $V$ is a tetrahedron whose vertices are $P(2,2,0), A(-2,0,0), B(0,0,2)$ and $C(1,1,3)$. I want to ask you how ...
0
votes
1answer
19 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 ...
1
vote
0answers
59 views

Calculating an integral with sine, cosine

I've recently calculated the Fourier transform of $\dfrac{\sin \pi ax}{\pi x}$. Now I'm trying to calculate $$\int _{\mathbb{R}} \frac{\sin ^2 \pi ax}{\pi ^2 x^3} \cos \pi bx\;\mathrm dx$$ The ...
1
vote
0answers
21 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
votes
1answer
54 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
votes
2answers
61 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
vote
0answers
29 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
95 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
votes
2answers
85 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 ...
0
votes
2answers
36 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
votes
1answer
36 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
votes
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
21 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 ...
1
vote
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 $$\lim_{y\rightarrow \infty}\int_0^1yx^yf(x)dx$$
0
votes
0answers
20 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 ...
0
votes
0answers
14 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 ...
0
votes
0answers
40 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
votes
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
43 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, ...
1
vote
2answers
72 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
votes
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
votes
5answers
234 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
vote
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
votes
1answer
60 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
votes
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
vote
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
votes
4answers
158 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
votes
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
vote
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
votes
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
vote
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
votes
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 ...
0
votes
1answer
34 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
votes
1answer
61 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
votes
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
votes
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
votes
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
votes
0answers
49 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 ...
0
votes
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 $\int_{0}^{\infty}x^{9}e^{-x^{2}} dx$ ?
1
vote
2answers
76 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
votes
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: $$\oint\frac{ds}{C} = \frac{2\pi r}{C}$$ ...
-1
votes
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 ...