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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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2answers
27 views

Integral of: $y^{2}.\exp(-(y^{2})/2)$

I need to calculate this integral of $\int_{0}^{\infty} y^{2} \exp(-y^{2}/2) dy$ I started doing by parts: $u = y$, $v= - \exp(-y^{2}/2)$, $du=dv$ and $dv=-y \exp(- y^{2}/2)$ So, $-y \exp(-y^{2}/2) +...
0
votes
3answers
46 views

Calculating improper integral $\displaystyle \int_0^1 \dfrac{\arcsin(x)}{\sqrt{1-x^2}}\,dx$

Calculate improper integral $\displaystyle \int_0^1\dfrac{\arcsin(x)}{\sqrt{1-x^2}}\,dx$ We had the following equation to calculate improper integrals (2nd style): Let f in$\left(a,b\right]$ ...
-1
votes
1answer
27 views

Evaluating the triple integral $\iiint (x-2) \,dx\,dy\,dz$ over a region

Here's the question $$\iiint (x-2) \,dx\,dy\,dz.$$ I am asked to evaluate this integral over the region $$D:=\left \{ (x,y,z) \in\mathbb{R}^3 :1\leq x^2+y^2+z^2 \leq9,x\leq z,z \geq 0\right \}.$$ I ...
2
votes
2answers
56 views

Integrating $\int_{0}^{a}\sqrt{{\tanh}(a)-{\tanh}(x)}\;dx$

What is the integral $$\int_{0}^{a}\sqrt{{\tanh}(a)-{\tanh}(x)}\;dx$$ I have tried various substitutions but couldn't get the answer. The substitution $x = {\tanh}^{-1}y$ has simplified into ...
0
votes
0answers
31 views

Definite integration when denominator consists of $x\sin x +1$

$$\int_0^\pi\frac{x^2\cos^2x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx$$ The answer is $0$. I tried and made $(x\sin x +1)^2 $ in numerator and proceed, but not able to do any further.
0
votes
0answers
31 views

Computing the arc length of the graph $y=\sqrt{x-x^2}+\arcsin(\sqrt{x})$

Computing the arc length of the graph $y=\sqrt{x-x^2}+\arcsin(\sqrt{x})$ Is this done right? I found the interval of this function which is $[0,1]$ I know the arc length formula is $$ L=\int_a^b\...
3
votes
1answer
57 views

$\displaystyle\int_{0}^{\infty}\frac{e^{-Ak^{2}}}{k}\sin(kr)dk$

I have the following integral $$ f(r)=\int_{0}^{\infty}\frac{\exp(-Ak^{2})}{k}\,\sin(kr)\,\mathrm{d}k $$ with $A>0$ and $r>0$. I know from Wolfram that the result should be $$ f(r)=\frac{\pi}{...
0
votes
1answer
16 views

Please explain how to take limits in double integral while finding volume using the given problem

Question: Find the volume under the surface $z=\sqrt{1-x^2}$ and above the triangle formed by $y=x$, $x=1$ and $x$ axis. The two integrals are given as follows: $$\int_0^1 \int_y^1 \sqrt {1-x^2} \,...
10
votes
5answers
183 views

Evaluating $\int_0^{\pi/2}\operatorname{arcsinh}(2\tan x)dx$

How to prove $$\int_0^{\pi/2}\operatorname{arcsinh}(2\tan x)dx=\frac43G+\frac13\pi\ln\left(2+\sqrt3\right),$$where $G$ is Catalan's constant? I have a premonition that this integral is related to $\...
2
votes
2answers
98 views

$\int\limits_0^\infty {x^4 \over (x^4-x^2+1)^4}\ dx$

I want to calculate $$\int\limits_0^\infty \frac{x^4}{(x^4-x^2+1)^4}dx$$ I have searched with keywords "\frac{x^4}{(x^4-x^2+1)^4}" and "x^4/(x^4-x^2+1)^4". But there are no results
1
vote
3answers
49 views

Help explaining the simplification of an integral

I am trying to understand the following steps my teacher did in class (from top to bottom). I tried to look up different trigonometric identities but couldn't figure out where the arrival of cosine ...
0
votes
1answer
39 views

product of $2$ definite integrals

If $\displaystyle I=\int^{\infty}_{0}e^{-2x}\cdot x^6dx$ and $\displaystyle J=\int^{2}_{0}x(8-x^3)^{\frac{1}{3}}dx$ Then product of $I$ and $J$ equals Try: For $\displaystyle I = \int^{\infty}_{...
5
votes
1answer
59 views

Definite Integration ( a little query)

$$\int_0^π \frac{xdx}{a^2\cos^2x+b^2\sin^2x} \,dx$$ Using property $$\int_a^b f(x) \,dx= \int_a^b f(a+b-x) \,dx$$ (i can't write it correctly,please check it) I get, $2I=\pi\int_0^\pi \frac{dx}{a^2\...
2
votes
0answers
52 views

In what sense does integration “raise the power by 1”

I am working on a problem that requires me to compare the value of a particular, but generically defined function, with the definite integral of that function. Naturally, if the function is a ...
3
votes
1answer
37 views

Degree choice in improper integrals resulting in trigonometric functions

people. I have a question regarding the following improper integral, and others like it: $$\int_{-\infty}^\infty \frac{dx}{1+x^2}$$ The end result of that are the two limits: $$\lim_{a\to -\infty} \...
0
votes
0answers
32 views

calculate this triple integral with $f(x,y,z)=(x^2+y^2)e^z$

I need to evaluate on this region : $D=\{1\le z \le 2,x^2+y^2 \le z^2 \}$ on this function : $f(x,y,z)=(x^2+y^2)e^z$ So the triple integral should be : $\int_{0}^{2\pi}\int_{1}^{2}\int_{0}^{z}r^3e^...
1
vote
1answer
22 views

Calculate flux Triple Integral

$R=\{z^2-4z+y^2 \le 0,0 \le x \le 1\}$ with $F=(x\sqrt{y^2+z^2},-z,y)$ So it's a shifted cylinder : $(z-2)^2+y^2=4$ $$ \left\{ \begin{aligned} x&=x\\ y&=2\sin\theta\\ z&=2+2\cos\theta \...
0
votes
1answer
19 views

How to find limits of this volume integration?

The question is If $\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }$ then find the value of $\int \int _ { S } \vec { F } \cdot \hat { n } d s$ where $S$ is the sphere $\{(x,y,z)\in\mathbb{...
0
votes
0answers
11 views

Function smoothness

why is this integrand non-smooth in +/- 1? ∫ √‾1‾- t^2‾ f(t) dt where f possesses an analytic extinction to a complex neighbourhood [-1,1] More general, how can I find it out? Thanks a lot!
1
vote
1answer
21 views

Show using the formal, limited based definition of integral

I am currently trying to figure out the following: for $f(x) = 2x - 5$, I want to show that, using the formal, limited based definition of integral, $\int_{3}^{7}f(x) = 20$ (both the domain and ...
3
votes
3answers
81 views

Calculate $\int\limits_{-\infty}^{\infty}\frac{\sin^2(x)\cos(wx)}{x^2}dx$ using complex analysis technique

In a complex analysis test an exercise asks to calculate ($w \in \Bbb{R}$): $$\displaystyle\int_{-\infty}^{\infty}\frac{\sin^2(x)\cos(wx)}{x^2}dx$$ Of course I need to solve it with complex analysis ...
16
votes
2answers
1k views

Evaluating definite integrals using Fundamental Theorem of Calculus

Here is a statement of the second part of the Fundamental Theorem of Calculus (FTC2), from a well-known calculus text (James Stewart, Calculus, 4th ed): If $f$ is continuous on $[a,b]$, then $\...
1
vote
2answers
20 views

Computation of a double exponential integral

I want to understand the behavior of this integral $$ \int_0^x e^{-\frac{c}{t^2}} \frac{1}{t^5} e^{-\frac{c_2}{(x-t)^2}} \frac{1}{(x-t)^5} dt. $$ The ideal answer would be a way to explicitly compute ...
10
votes
1answer
178 views

Integral $\int_{\sqrt{33}}^\infty\frac{dx}{\sqrt{x^3-11x^2+11x+121}}$

How can we prove $$I:=\int_{\sqrt{33}}^\infty\frac{dx}{\sqrt{x^3-11x^2+11x+121}}\\=\frac1{6\sqrt2\pi^2}\Gamma(1/11)\Gamma(3/11)\Gamma(4/11)\Gamma(5/11)\Gamma(9/11)?$$ Thoughts of this integral This ...
-1
votes
0answers
30 views

The integral of x^x-a where a is a real number [closed]

How to calculate the integarl of x^(x-a) for all x runs from 1 up to t and a is a real number. Thank you!
1
vote
0answers
35 views

Integration by parts in proof of special case of Ehrenfest's theorem

Our physics professor proved a special case of Ehrenfest's theorem, or to be more precise: $$m\langle x\rangle = \langle p\rangle,$$ where we have used the definition: $$\langle x\rangle = \int_{\...
0
votes
0answers
42 views

There is c such that $f''(c)g(c)+f(c)g''(c)+f'(c)g'(c)=0$

Let $f,g:[a,b]\to\mathbb{R}$ both increasing, twice differentiable and $g$ is also convex, $f(a)=0$, $g(b)=0$, $f^\prime(a)=0$. Prove that there exists $c\in(a,b)$ such that $$f''(c)g(c)+f(c)g''(c)+f'(...
3
votes
1answer
60 views

Trying to solve this triple integral: $\iiint (x-1)(y-1) \,dx\,dy\,dz$

Here's the question $$\iiint (x-1)(y-1) \,dx\,dy\,dz.$$ I am asked to evaluate this integral over the region $$D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2+y^2 \leq z \leq 2x+2y+2 \right \}.$$ There are ...
2
votes
1answer
62 views

Solving a difficult differential equation

Well, I've to solve the following DE: $$y(t)=x(t)\cdot\text{a}+\text{b}\cdot\ln\left(1+\frac{x(t)}{\text{c}}\right)+\int_0^tx(\tau)\cdot p(t-\tau)\space\text{d}\tau\tag1$$ And I've no idea how to ...
0
votes
1answer
35 views

Integration of greatest integer function [closed]

Let $[x]$ denote the greatest integer function, then evaluate $$\int_{-100}^{100}[x^3]dx$$ I have no idea how to go about it, can someone help me out
0
votes
0answers
17 views

Integration over two functions

I need to integrate a function of the form: $H(t) = \int_0^t f(u)g(t-u) du$ The functions are fixed, but aren't simple and so I'm using numerical methods to calculate the value of $H$. However, I ...
0
votes
1answer
32 views

Arccotx definite integral

Need help with.the question. The solution is given but I have no idea how they have broken into intervals. Would love some help.
0
votes
2answers
38 views

Volume of solid using shell method

I have a homework question that asks the following: Use the shell method to find the volume of the solid generated by revolving the region bounded by the line $y=3x+4$ and the parabola $y=x^2$ ...
0
votes
1answer
56 views

How is this integral computed??

I'm reading this book about electrical properties of materials where the electron is introduced as a wave. Using the equation of a wave, they bring about the "envelope" of a wave. So here is how the ...
1
vote
1answer
34 views

evaluate this elliptic hyperbloid volume?

How can I calculate the volume of this region in cylindrical coordinates? $D=\{2x^2+y^2=z^2+4,|z| \le 2\}$ I think I got this wrong : $$\operatorname{Volume} = 2\int_{0}^{2\pi}\int_{0}^{2} \int_{0}^...
0
votes
0answers
15 views

Integrating function with substitution of variable

I have the following integral: $$ 1 = N^{-1}\int_0^{R_M}\bar n(R)4\pi R^2 dR = \frac{\sqrt{2}\pi}{3}\int_0^{R_M}g_s(R)R^{-1}dR $$ Note that $x=R/\sigma$ I would like to solve this integral and ...
1
vote
1answer
68 views
+50

Using cylindrical coordinates, find the volume of the region $D =\{y^2+z^2\le5+x^2,4x^2+y^2+z^2\le25\}$

I want to calculate this volume region, using cylindrical coordinates: $$D=\{y^2+z^2\le5+x^2,4x^2+y^2+z^2\le25\}$$ So, I have a hyperboloid and an ellipsoid. Is it correct to calculate the volume ...
0
votes
1answer
26 views

triple integral over a shifted cylinder

$D=\{(y−2)^2+x^2\le1,0≤z≤2\} $ I need to calculate the volume. so I'ts a cylinder with radius 1, and shifted by 2 units in the y axe. The problem here is that the circle doesn't touch the origin! so ...
3
votes
3answers
122 views

How would you calculate this limit? $\lim\limits_{n \rightarrow\infty}\frac{\pi}{2n}\sum\limits_{k=1}^{n}\cos\left(\frac{\pi}{2n}k\right)$

I decided to calculate $\int_{0}^{\pi/2}cos(x)dx$ using the sum definition of the integral. Obviously the answer is $1$ . I managed to calculate the resulting limit using the geometric series, taking ...
1
vote
1answer
28 views

Evaluate this triple integral (volume)

I need to calculate this volume of $$D=\{x^2+y^2-2y\le 0,0\le z\le 10-3\sqrt{x^2+y^2} \}.$$ My attempt. So the first one is a shifted cylinder $x^2+(y-1)^2\le 1$ , and the second one is an upside-...
0
votes
1answer
25 views

A question about integral inequality [closed]

$f$ is differentiable on $[-1,1]$, $M=\sup|f'|$. There is $a\in(0,1)$ such that $\int^a_{-a} f(x)dx=0$. Prove that $|\int^1_{-1} f(x)dx|\leq M(1-a^2)$
-2
votes
1answer
30 views

Finding region of double integral [closed]

How can i calculate the double integral of $x^2+y^2$ which is bounded between the lines $x=1 , x=2$ and $y=x , y=x^{1/2}$ ? Should i look at the region between the graphs of the above functions ?
4
votes
2answers
126 views

Is there a closed form for the trigonometric integral $\int\limits_0^{\pi/4}\frac{\cos(2k+1)x}{\cos x} dx$?

One can easily show that $\int\limits_0^{\pi}\frac{\cos(2k+1)x}{\cos x} dx = 2 \int\limits_0^{\frac{\pi}{2}}\frac{\cos(2k+1)x}{\cos x} dx = (-1)^k \pi$. But is there a closed form for $\int\limits_0^...
12
votes
3answers
249 views

Finding a closed form for $\int_{0}^{1}\frac{\ln\left ( 1-x^{2} \right )\arcsin ^{2}x}{x^{2}}\mathrm{d}x\approx -0.939332$ [duplicate]

I'm attempting to find a closed form for $$\int_{0}^{1}\frac{\ln\left ( 1-x^{2} \right )\arcsin ^{2}x}{x^{2}}\mathrm{d}x\approx -0.939332$$ I tried to use $$\displaystyle \arcsin^{2}x=\frac{1}{2}\...
3
votes
2answers
102 views

The closest value to $\int_0^1 \sqrt{1+\frac{1}{3x}} dx$?

This is a multiple choice among 1.6, 2, 1.2. So the approximation should be sufficiently accurate. The solution is 1.6 as can be verified using Taylor expansion. But Taylor expansion method takes too ...
-2
votes
0answers
19 views

Partial Derivative of Definite Integral

I am trying to find a way to partially derive the following equation with respect to each of its variables, C, BC, BS, and BG (First Order Conditions): (Integral from 0 to T) f(C,BC,BS,BG)
0
votes
1answer
54 views

Integral identity in statistical physics [closed]

I found this identity in a statistical physics book, but I can't find a way to demonstrate it and I don't like to take thing as true without knowing why. The identity is: $$ |z|=\frac{1}{\pi} \int_0^...
0
votes
1answer
47 views

How to prove that a function obeys a certain functional equation

I saw this equation in a book and It was used as part of calculating something else in here and here Can I get any help about this equation the statement is this: suppose that: $$f: R\times R \...
-3
votes
0answers
31 views

Simpson integral formula [closed]

Does anyone know how to solve integral $\ln x/(1+(\sin x)^2)$ in boundaries from 1 to 4, with precision $0.5\cdot10^{-3}$. I have a problem solving 4th derivate from this function...
0
votes
1answer
28 views

proving definite integral value by induction

I have been trying to prove the following by induction but all my efforts have failed. the original problem is to prove that $\int_{-1}^{1}(1-x^{2})^kdx = \frac{2^{2k+1}(k!)^{2}}{(2k+1)!}$ Where $...