Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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7
votes
2answers
150 views

Test for convergence $\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$

What is the easiest way to test the convergence of $$\int_0^{\infty} \frac{\sin(x)}{x+\log(x)} \ dx$$ Is it possible to only use the high school tools for that?
0
votes
1answer
30 views

what is the order of integration for : integral of x dx * integral of y dy

I'm still trying to get my head around he basics of this stuff so please use simple language in your answer ! $$ \int dx \int f(x,y) dy$$ the first integral limits are from 0 to 1 for dx and the ...
0
votes
0answers
54 views

Is it possible to solve this set of equations?

Let's have system of equations: $$ \tag 1 [\nabla \times \mathbf E ] = -\frac{\partial \mathbf B}{\partial t} , $$ $$ \tag 2 [\nabla \times \mathbf B] = \sigma \mathbf E + A(\mu \mathbf K + C \mathbf ...
0
votes
0answers
40 views

How to solve the following integral [closed]

Do you have any idea how to solve the following integral: $$\int\limits_0^a {{e^{\large \left(- \frac{{by}}{{c - dy}} - ey\right)}}dy}$$ where $a$, $b$, $c$, $d$ and $e$ are constants? Thank you ...
15
votes
6answers
402 views

Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$

How to evaluate the following integral $$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$ It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of ...
0
votes
1answer
63 views

If a continuous function on $[0,\pi]$ integrates to zero against cosines, it is identically constant

Let $f(x)$ is Continuous function on $[0,\pi]$,and for $n=1,2,.....,$ the function $f(x)$ has the following property:$$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0.(n=1,2,......)$$ Proof: $f(x)\equiv C$(C is ...
2
votes
2answers
140 views

How to show that $\int_0^1 dx \frac{1+x^a}{(1+x)^{a+2}} = \frac{1}{a+1}$?

From numerical evidence it appears that whenever the integral converges, $$J_a :=\int_0^1 dx \frac{1+x^a}{(1+x)^{a+2}} = \frac{1}{a+1}.$$ For $a \in \mathbb{N}$, I was able to prove this using ...
1
vote
0answers
42 views

Accelerometer data integration (MMSE)

Based on the raw accelerometer measurements use simple integration on the raw $X$ and $Y$ axis data to determine the velocity and position. If we assume a linear model $Y = aX + b$ for determine the ...
1
vote
2answers
49 views

Integral of pdf

I need to find the integral for this pdf but I don't know if I need to, or can, take the integral of two variables at the same time. $$ f(x;\theta)=\frac{x}{\theta^2} e^{-x^2/(2\theta^2)} ,\quad ...
2
votes
1answer
57 views

If functions converge a.e. and their integrals converge, does convergence in $L^1$ follow?

I was wondering if $f_n, f:\mathbb{R}\rightarrow\mathbb{R}$ are s.t. $f_n\rightarrow f$ pointwise a.e. and $\int f_n\rightarrow \int f$ where integrals are Lebesgue Integrals, is there any Theorem or ...
0
votes
3answers
66 views

how to integrate $\int_{0}^1 \sqrt{(e^x+e^{-x}+2)} dx $? [closed]

what to find $\int_{0}^1 \sqrt{(e^x+e^{-x}+2)} dx $ ? Could you give me a hint? Thanks!
1
vote
2answers
77 views

Integration with square root in denominator

I am honestly embarrassed to ask this because i feel like i should know how to do this but: $ \int \frac{x}{\sqrt{2x-1}}dx $ Try to use u-substitution please
0
votes
2answers
37 views

Gamma function in $C^{2}$

How can I show that for $x>0$, the Gamma function is at least $C^{2}$? The Gamma function is defined as $$\displaystyle \int^\infty_0 e^{-t}t^{x-1}\ dt$$ For which $x$ is the integrand integrable?
1
vote
2answers
108 views

How to integrate $\int \frac{\sqrt{x}}{x+1}dx$?

How to integrate $$\int \frac{\sqrt{x}}{x+1}dx$$ Can I substitute $x+1$ with $u$?
1
vote
1answer
56 views

Solving the integral which shows the second moment of subtracting two Beta-distributed Random Variables

Peace be upon you In my project I needed to find the second moment of the subtraction of two Beta-distributed random variables. I have computed it and reached to the following integral which I should ...
0
votes
1answer
88 views

Fubini's theorem, interchanging order of integration

My question is, imagine I want to compute the following integral: $$\int_A \int_B f(x,y)dxdy$$ and I decide to start from $x$ and get $$\int_A \int_B f(x,y) dxdy <\infty.$$ On the contrary if I ...
2
votes
0answers
52 views

Calculating the Integral of a non conservative vector field

I have no clue how to do part C because a) is non conservative What I got for b) $f(x,y)=\frac{x^3}{3}+2yx+\frac{y^3}{3}+K$ (I don't know the symbol for the thing so I used f(x,y) instead. How do I ...
0
votes
0answers
19 views

Estimates for the wave equation

Spose $ u $ solves the wave equation on $ U \subset \mathbb{R}^3 $ with initial conditions $ u (x, 0) = g(x)$ and $ u_t(x,0) = h(x)$, where lower script indicates partial differentiation. Then we have ...
2
votes
2answers
211 views

Integrating the gamma function

I assumed that $$\Gamma\left(k+\frac{1}{2}\right)=2\int^\infty_0 e^{-x^2}x^{2k}\,dx=\frac{\sqrt{\pi}(2k)!}{4^k k!} \,,\space k>-\frac{1}{2}$$ and that ...
5
votes
0answers
60 views

How do I do this change of variables?

Use a change of variables to evaluate: $$\iiint\limits_{D}xy\,\mathrm{d}V$$$D$ is bounded by the planes $y-x=0$, $y-x = 2$, $z-y = 0$, $z-y = 1$, $z=0$, $z=3$. I set $$u = y-x$$ $$v = z-y$$ $$w ...
3
votes
1answer
62 views

A suitable integration path for $\cos z/(2 + \cos z)$

I was solving the exercises and got stuck when trying to solve this with tools of residual calculus $$ \int_{0}^{2 \pi} \frac{\cos (z)}{2 + \cos (z)} \, dz = \int_{0}^{2 \pi} f(z) \, dz. $$ Isolated ...
1
vote
2answers
56 views

Calculus: volume of revolution about a line other than the $x$-axis.

Find the volume of the solid of revolution obtained by rotating the region bounded by $f(x) = x^3 + 1$, $g(x) = x^2$ and $0 ≤ x ≤ 1$ about the line $y = 3$. I know the gist of the problem, but ...
-1
votes
2answers
45 views

Calculate line integral

For $f(x,y) = 2x + y + 10$, calculate the line integral $$ \int_{L}{f(x,y)dL} $$ where $L$ is the straight line between $(1,4)$ and $(5,1)$ in the $xy$-plane.
6
votes
3answers
109 views

Integration $I_n=\int_{0}^{1}\frac{dx}{(x^n+1)(\sqrt[n]{x^n+1})}$

$$I_n=\int_{0}^{1}\frac{dx}{(x^n+1)\large\sqrt[n]{\normalsize x^n+1}}$$ Could someone help me through this problem?
0
votes
1answer
56 views

Integration by Substitution

Question: Use the substitution $u=\tan x$ to find $\displaystyle \int_{0}^{\large \frac{\pi}{4}}\left(\tan^{n+2}x + \tan^{n}x\right)dx$. Using the above result, find the exact value of ...
2
votes
0answers
39 views

How to relate two integration contour?

How one can relate two integration contour? For example if I have an integration contour like $\int_{-a}^{a}f(x)dx$ here let say a=infinity. How I can say that the integral $\int_{2}^{3}f(x)dx$ is a ...
1
vote
1answer
30 views

Determine integral by using the following identity (which is imaginairy)?

I want to determine the following integral: $$\int_{-\infty}^\infty \frac1{x^6+1} dx$$ by using the following identity: $$\frac1{x^6+1} = \Im\left[\frac1{x^3-i}\right]$$ How in the world can I do ...
0
votes
3answers
55 views

Integration separation of variable

Liquid is flowing into a small tank which has a leak. Initially the tank is empty and, t minutes later, the volume of liquid in the tank is V cm3 . The liquid is flowing into the tank at a constant ...
7
votes
1answer
178 views

Closed-form of $\int_0^1 \int_0^1 \int_0^1 x^{(y^z)} \,dz\,dy\,dx$

We know that $$\int_0^1 \int_0^1 x^y\,dy\,dx = \ln 2.$$ Do we know a closed-form of $$\int_0^1 \int_0^1 \int_0^1 x^{(y^z)} \,dz\,dy\,dx\,?$$ As a start we know that $$\int_0^1 x^{(y^z)}\,dz = ...
0
votes
2answers
66 views

By applying the second version of the Fundamental Theorem of Calculus find the integral:

The second version of the Fundamental Theorem of Calculus states that if $F'(x)=f(x)$ then $\int_{a}^{b} f(x) dx = F(b)-F(a)$. I need to use this to find a) $\int_{-2}^{-1} \frac{1}{x^3} dx $ and b) ...
4
votes
1answer
74 views

Integrate $e^{-\frac{y^2}{2}}\left(\frac{1}{y^2}+1\right)$

I'm trying to find $$\displaystyle \int{e^{-\frac{y^2}{2}}} \left(\frac{1}{y^2}+1\right)dy$$ I tried using integration by parts and some substitutions, but nothing seem to work. The answer is ...
2
votes
2answers
79 views

Evaluate $\int\sec^4(u) \operatorname d \!u$

Evaluate $$\int\sec^4(u) \operatorname d \!u$$ I don't know what to substitute: I've tried $1+\tan(u)$ and integration by parts. I know the general formula for $\sec^n(u)$, but I want to be able to ...
2
votes
2answers
193 views

How to evaluate $\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x$?

How to evaluate the following integral $$\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x $$ The numerical result is $= -1.38104$ and when I look at it, I have no idea how to ...
3
votes
2answers
58 views

how can i change specifically the intervals of a double integral?

I know how to change the intervals of an integral, for example the integral of $(\sin x)^2$ from $-\pi$ to $\pi$ is equal to $\pi\int_{-1}^1 (\sin πx)^2 \,dx$. I find it difficult to do that in 2D. ...
0
votes
0answers
43 views

Is there a way to use this interpretation of differential forms on manifolds?

I read Rudin's "Principles of Mathematical Analysis". In the part of Differential Forms, he defined them formally. I particularly enjoyed the formal viewpoint, since everywhere else it seems that the ...
3
votes
0answers
46 views

How to find the Maclaurin series for the integral of $e^{x^2}$?

I am trying to find the Maclaurin series for the integral of $e^{x^2}$? What I done so far is that the Maclaurin series for $e^{x^2}$ is $$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$$ So would ...
12
votes
1answer
174 views

$\int\limits_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int\limits_{f(a)}^{f(b)} f^{-1}(x) dx$ proof

I just wanted to ask, if my proof is correct. I haven't seen the equation before, but I think it's quite useful. Let $f$ be an bijective differentiable function. Then the inverse function $f^{-1}$ ...
3
votes
2answers
66 views

Convergence of sequence of integrals.

Let $(\mathcal{X}, \mathcal{A}, \mu)$ be a measure space, $f_n: \mathcal{X} \to \Bbb R$ a sequence of measurable functions, and $g_n:\mathcal{X} \to \Bbb R$ integrable functions such that $|f_n| ...
4
votes
2answers
58 views

Any idea ? $\int {\sqrt{1+\sqrt{x}}}/x dx$

$\int \frac{{\sqrt{1+\sqrt{x}}}}{x} dx$ I try with $u=\sqrt x $ but i don't know what to do... Thank you Shadock
1
vote
1answer
25 views

how to show this manipulation in the integral

Let we have: $$G(t)=y_1(t)\int y_2(s)ds$$ when we take the limits as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds$$ then is it possible to write it as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds=\int^t_{t_0} ...
0
votes
1answer
54 views

Center of mass of a barrel partially filled with grain

Bubba has a barrel in the shape of a cylinder of mass 39.4 kg. The barrel has a diameter of 62.4 cm and is 1.32 m tall. He fills the barrel to a depth of 49.5 cm with loose packed grain that has an ...
0
votes
0answers
60 views

Triple integral containing definite integral and exponentials with trigonometric functions

I am attempting to solve the following integral analytically: $$ \int_{z=5i}^{z=1} \int_{t=\csc^{-12}(z)}^{t=2} \int_{\theta=\sin^{t}(z)}^{\theta=t^2} {[\mathrm{e}^{t\cos(\mathrm{e}^{i \theta})} + ...
0
votes
3answers
21 views

Integrate the differential equation of a simple rate equation

Could somebody please show me how to integrate the following: $dA/dt = -kA$ I'm told that the answer is: $A(t) = A(0)e^-kt$ but I do not know why. Could you be explicit in your answer and explain ...
0
votes
1answer
51 views

Calculating a limit of integrals

I am having a problem with the following exercise: Show that for every bounded borelian function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$, $\underset{n}{lim} \frac{n}{\sqrt{2\pi}} ...
1
vote
1answer
47 views

Equivalent form of a double integral.

I am looking at the second question of this problem set: The iterated integral $\int_0^1 \int_{y/2}^1 e^{x^2} dx \, dy$ can be expressed as (a) $\int_0^1 \int_0^{2x} e^{x^2} dy \, dx$ ...
0
votes
2answers
70 views

Fundamental theorem of calculus problem - trig functions

My problem is: On the interval (0 , pi/2). I know I need to split it in two integrals, but I don't know how. I would appreciate any suggestions on how to proceed.
0
votes
2answers
41 views

Integral with parameter: $\int_{0}^{a}x^2\sqrt{a^2-x^2}dx$

I have the following integral : $$\int_{0}^{a}x^2\sqrt{a^2-x^2}dx$$ I tried to manipulate the integral and then use substitution to get a rational form to arrive at: $$-8a^4\int_0^a ...
1
vote
0answers
46 views

Integration question with square root

$\int\sqrt {(1+3x^2+6x^3}dx$ I tried taking the substitution $u^2=1+3x^2+6x^3$. I was able to simplify the integral to $\frac{1}{(3x)(1+3x)} + \frac{x}{1+3x} + \frac{2x^2}{1+3x}$. I know I can form ...
0
votes
1answer
60 views

Proving $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)} dx < \frac{1}{e}$

While i was playing around with very weird functions and came across this: $$ \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)}dx \approx 0.3676932086...\approx \frac{1}{e} - ...
1
vote
0answers
34 views

Does there exist such function?

Fix an integer value $k\geq 1$. Let $[0,1]$ the unit interval and let $s\in [0,1]$. Does there exist a function $f$ (which depends on $k$ of course but not on $s$) such that $$\int_s^1 \left( ...