0
votes
0answers
21 views

A question about properties of integrals

Suppose g is differentiable with $g'(x)<0$ for all $x<1$, and $g'(x)>0$ for all $x>1$, and suppose $g(1)=0$. Now let $G(x)=\int_0^x g(t)dt$. Prove that G(x) is an increasing function (this ...
0
votes
0answers
16 views

Requirements for integration by parts

In order to use the integration by parts formula for functions of several variables $$\int_{\Omega} \nabla u\cdot v d \Omega = \int_{\partial \Omega}(u(v \cdot \nu))d \Omega - \int_{\Omega}u\nabla ...
0
votes
0answers
14 views

Integration on Manifold

I am beginning my studies on integration on manifolds and i have some theorical questions. First, in all books that I saw they says that the singular p - simplex (or p - cube) are continuous mapping ...
1
vote
0answers
335 views

Is $ \lim_{x \to 0} G(e^x) x = \sum_{r=1}^\infty \mu(r) \int_{-\infty}^\infty \frac{G(z^r) dz}{z} $?

I recently derived a formula and was wondering if it was correct or already existing? We define: $$ \sum_{r=1}^\infty \mu(r) G(x^r) = g(x) $$ Where $ \mu(r) $ is the Mobius function. G(x) is an ...
1
vote
1answer
54 views

How to calculate an integral

I wonder how the integral $$\int_{-1}^1 \! \int_0^{\sqrt{1-x^2}} \! \int_0^{\sqrt{1-y^2-x^2}} \! 1 \, dz \, dy \, dx $$ Any ideas?
3
votes
1answer
44 views

Integration to Gamma Function?

I need to show that $$\int\limits_{0}^{\infty}\theta^{-\tau(\alpha_1 + \alpha_2) - 1}e^{-\theta^{-\tau}\left(y^{\tau} + \delta^{\tau}\right)}\text{ d}\theta = \dfrac{\Gamma\left(\alpha_1 + ...
0
votes
1answer
42 views

Calculus, find the area between two given functions

I wonder why my answers were wrong, I equaled the two functions and set them equal to zero. then I found the integral and substitute with the the given points. ex: $cosx - e^x$ integration of ...
0
votes
1answer
8 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
4
votes
6answers
95 views

If $\lim\limits_{x \to \infty} f(x) = 1$, can we have function $f(x)$, such that $\int_0^{\infty}f(x)dx$ converges

I know the Initiative answer, can anyone give a neat answer based on solid reasoning EDIT : $f(x)$ is continuous
0
votes
4answers
127 views
0
votes
2answers
17 views

Line integrals; How to set $t$ boundary?

I'm having a hard time understanding how to set t boundaries in line integrals... The question is: find the line integral of $f(x,y,z)$ over the straight line segment from $(1,2,3)$ to $(0,-1,1)$. I ...
0
votes
0answers
18 views

integration coordinates

Could anyone give me hint on how to do it? I know that I have to find the y values by: F(b)= F(a) + a-b integral f(x) dx F(b) = 150 + a-b integral f(x) dx but how to find the integral from 0 to ...
2
votes
0answers
64 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting to rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} ...
-1
votes
0answers
43 views

Evaluating $\int\frac{1}{(x^2-5)^{0.5}}\,d(x^2+5).$ [on hold]

How can I evaluate $$\int\frac{1}{(x^2-5)^{0.5}}\,d(x^2+5)?$$ Thanks in advance!
0
votes
1answer
44 views

How can the signed area be 0?

How can the signed area be 0? For example if you have 3 on positive x side and 3 on the negative x side then you get the signed area of 0? How can area be 0?
1
vote
1answer
22 views

Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
1
vote
0answers
58 views

Can there be a power series with interval of convergence $[k, \infty)$?

My answer : NO Because Interval of convergence is of the form $(a-R, a+R)$ Where $a$ is centre of convergence. If there exists a power series with Interval of convergence $[k, \infty)$ $ $ We ...
0
votes
0answers
29 views

Asymptotic analysis if t tends to infinity [on hold]

Asymptotic analysis if t is large. p=1 is making contribution to the asymptotic behavior?
2
votes
3answers
84 views

Find $x > 0$ for which $\int_{0}^{x} [t]^2 \ dt = 2 (x-1)$.

What are all possible $x > 0$ for which the following equation is satisfied? $$\int_{0}^{x} [t]^2 \ dt = 2 (x-1),$$ where $[.]$ denotes the bracket (or floor) function. I guess we will have to ...
5
votes
2answers
137 views

Prove that $f$ is constant on $[a,b]$

$\displaystyle \int_{a}^{b} f^2(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^4(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^3(x) \, \mathrm{d}x$ And $f$ is continious on $[a,b]$ and ...
1
vote
0answers
31 views

Error bound by the Simpson's rule

My lecture notes have a little exercise. Two functions are given: $$ f(x) = \cos(x) \ \text{and} \ g(x)=\sqrt{x+1} $$ And we're asked about the error bound of the Simpson's rule to estimate the ...
2
votes
2answers
64 views

When not to use integration by parts?

I am trying to evaluate this integral using integration by parts. $$I=\int_{0}^{\infty}f(x)g'(x)dx,$$ where $f(x)=\sin x$ and $g'(x)=\dfrac{x}{1+x^2}$. So: $f'(x)=\cos x$ and ...
0
votes
0answers
36 views
+50

$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$

The following integral bothers me since weeks: $$ \int_0^\infty (1+t^2)^{-s} (1+it)^{s'} 2t \; d t.$$ Has any body a suggestion for this integral. $Re\; s >0$ sufficiently large and $s'$ an ...
1
vote
1answer
31 views

consider a square of side length $x$, find the area of the region which contains the points which are closer to its centre than the sides.

Any ideas how to start. I am having trouble figuring out the region itself All ideas are appreciated thanks
1
vote
1answer
34 views

Moving $d$ terms inside a double integral?

I was dealing with an integral expression like that: $$\int zf(z)dz$$ In this term it is known that $f(z)=\int g(x,z) dx$. So I can replace $f(z)$ in the first term like that: $$\int z(\int ...
1
vote
1answer
41 views
0
votes
3answers
53 views

Find $\int_0^4\int_{0}^{4}xy \sqrt{1+x^2+y^2} \,dy\, dx $

I am having a tough time figuring this one out. Any help will be appreciated. do we have to approximate, or can we actually find it
1
vote
1answer
36 views

Volumes of Revolution Washer Method

I have to find the volume of revolution of a region called $C$ using around the $y=-1$ axis. The region is bounded above by $y \ = \ \ln(x+1)$, bounded below by $y=e^{-x}$ and on the right by $x=3$. ...
1
vote
4answers
314 views

Why does the difference between the left-hand and right-hand sum get smaller with more subdivisions?

Q) For a given function on a given interval, the difference between the left-hand sum and right-hand sum always gets smaller as the number of subdivisions gets larger. I remember that the answer is ...
1
vote
1answer
34 views

True or False and why ? basic integration [duplicate]

$$\int_0^2f(x)\,\mathrm dx=\int_0^2 f(t)\,\mathrm dt.$$ I would say true because a&b = a&b and it would not make a change if is x,d,f,q,w,t it will logically be all the same, am I right ? but ...
0
votes
1answer
34 views

Calculus about basic integrals

I have to find the lower and upper estimation. ( Hint = you should sketch a possible graph and draw the rectangles.) My answer for the upper estimation was : 2( 23+15+6+33) = 154 and My answer for ...
3
votes
4answers
170 views

Integrals piecewise / basics

Consider the following function: $$f(x)=\begin{cases} 7-x, &\: 0 \leqslant x \leqslant 7 \\ x-7, &\: 7 \lt x \leqslant 14 \end{cases}.$$ Find the exact value ...
0
votes
1answer
13 views

Integral, left-hand sum

Could anyone explain why my first answer is wrong? what I did was delta x = 10/5 = 2 $$ 2(2^2+1)+2(4^2+1)+2(6^2+1)+2(8^2+1) = 248 $$ and the second answer was $$ ...
2
votes
4answers
109 views

Fastest way to integrate $\int_0^1 x^{2}\sqrt{x+x^2} \hspace{2mm}dx $

This integral looks simple, but it appears that its not so. All Ideas are welcome, no Idea is bad, it may not work in this problem, but may be useful in some other case some other day ! :)
4
votes
4answers
275 views

Find the Value of Integral

Find the Value of $$\begin{align}I=\int_{0}^{1}\frac{\ln(x)\,dx}{1-x^2}\end{align}$$ I have tried like this: We have ...
4
votes
2answers
92 views

Compute integral.$\int_0^{\pi/2}log(a^2sin^2x+b^2cos^2 x )~dx$

This is a integral for a calculus exam, and I have no idea how to solve it. $$\int_0^{\frac{\pi}{2}} \log \big( a^2 \sin^{2}(x)+b^2 \cos^{2}(x) \big) \, \mathrm{d}x$$
1
vote
1answer
42 views

How to evaluate this integral?

How to evaluate $$ \int_{a}^{b} [x] \ dx \ \ + \int_{a}^{b} [-x] \ dx \ ? $$ I know that $[-x] = -[x]$ if $x$ is an integer, whereas $[-x] = -[x] - 1$ if $x$ is not an integer. So is it about ...
0
votes
1answer
52 views

Evaluating $\dfrac{1}{\Gamma (r)}\int_{0}^{x}(x-t)^{\alpha -1}t^{\lambda}dt$ [closed]

How can I evaluate the following integral $$\frac1{\Gamma(r)}\int_0^x(x-t)^{\alpha-1}t^\lambda\ dt$$
0
votes
1answer
37 views

Changing order of integration in cylindrical coordinates

I'm having a problem in changing order of integration in triple integration, in cylindrical coordinates. I would be grateful for a little help.The question is: Let D be the region bounded below by ...
1
vote
3answers
364 views

The high power integral

Im trying to solve the indefinite integral $$\int\frac{x}{(x^2+4)^3} \, \mathrm{d}x $$ I tried applying polynimial division and breaking to partial fractions but it didnt help...are there any other ...
6
votes
3answers
443 views

Are indefinite integrals unique up to the constant of integration?

We often write e.g. $$\int x^2 dx=\tfrac{1}{3}x^3+c$$ for any $c \in \mathbb{R}$, where $c$ is the constant of integration. We can show (via limits) that, if $g(x)=\frac{1}{3}x^3+c$, then ...
2
votes
0answers
53 views

Simplifying a Vector Integral

While reading the book - Cercignani, Theory and Applications of Boltzmann Transport Equation (I am not a math student), I found this integral which I am unable to understand. Note that $\xi_i , \xi_l$ ...
0
votes
1answer
41 views

Integral with square root of function of function

I have the function $y=y(x)$ with $y'=dy/dx$, and the following equation: $ky'=\pm\sqrt{k^{2}-y^{2}}$, where $k$ is constant. Integrating this, given that $y(0)=0$, should give: $y=k\sin(x/k)$. I ...
0
votes
0answers
29 views

Integration Help Needed to Calculate Arc Length of a Vector-Valued Function

Consider the path of a particle in a conservative force field represented by the vector-valued function r(t)=(4(sint−tcost),4(sint+tsint),32t2). Find the arc length function s. I have calculated ...
1
vote
3answers
98 views

Integrate $\tan^4(\theta)$

I know the answer but I don't know how to find the proper u-sub for it. I'm told I need to use U-sub for all the integrals. Here is where I end up $$\int (1-\sec^2(\theta))\,d\theta -\int ...
0
votes
1answer
59 views

$\int_0^t m(x) \ dx =\int_0^t n(x) \ dx $ for all t > 0 implies that m(x) = n(x)

$$\int_0^t m(x) \ dx =\int_0^t n(x) \mathrm{d}x $$ For all $t > 0$. m and n are continuous. Prove that $m(x)=n(x)$ MY APPROACH I took f(x)= m(x)-n(x). Assume that f(x) is not zero at ...
-1
votes
3answers
74 views

Can there be more than one power series expansion for a function.

I guess the answer is NO, for polynomials. I know that there are more than one series expansion for every function. But I am talking about power series here. All Ideas are appreciated
1
vote
1answer
59 views

Proving $\int^\infty_0 x^n e^{-x} \, dx = n!$

I was motivated by this question on the various applications of integration by parts to prove the following integral: $$\int^\infty_0 x^n e^{-x} \, dx = n!$$ Here's what I have done, I feel I am ...
0
votes
1answer
24 views

Radius of curvature and continuous functions

Let $\kappa (x)$ be radius of curvature function for a continuous function $f(x)$. Is it necessary that $\kappa(x)$ will have extrema when $f(x)$ does. And the nature of extrema is opposite to that ...
4
votes
1answer
84 views

Solving for limit of integration

$$\frac{1}{\sqrt{2\pi}} \int^0_{z_a} e^{\frac{-z^2}{2}} \, dz = 0.48 $$ How would I solve for the value of $z_a$ using a calculator?