All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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1answer
126 views

Volume of the region?

Hello I'm trying to solve this equation but kinda stuck. The problem is as follows; The region in the first quadrant bounded above by the line y =2, below by the curve y=2sinx, $0\le ...
2
votes
3answers
61 views

Integral with log x calculations

The integral is: $$\int_1^e x^2\log(x)\,dx$$ I don't know what to do with $\log$
2
votes
1answer
43 views

How can I find this integral?

I want to find the integral $$\int{\dfrac{\cos^2 x}{\sin^2 x + 4\sin x \cos x}} \mathrm{d} x.$$ I tried put $t=\tan \frac{x}{2}$, I got $$\int{-\dfrac{1}{4}\dfrac{(t^2-1)^2}{t(t^2 + 1)(2t^2-t-2)} ...
5
votes
5answers
209 views

Why isn't there a fixed procedure to find the integral of a function? [duplicate]

Since the integration of a function is the opposite of a the derivative of a function, and there are clear steps to follow when we want to find the derivative of a function, I thought there would be ...
4
votes
1answer
125 views

Analytical solution of integral $x^{1+n}\exp(-x) \sin(x)$

$$\int\limits_0^\infty x^{1+n} \exp(-x) \sin(xy) \mathrm{d}x $$ for real $n, x, y$ and $n\ge-3$ has the following analytic solution according to mathematica: $$(1 + y^2)^{\left(-1 - ...
0
votes
2answers
146 views

Finding the integral of $\left[3x\sin\left(\dfrac x4\right)\right]$.

$\displaystyle\color{darkblue}{3\int x\sin\left(\dfrac x4\right)\,\mathrm dx}$ $$\begin{align} \dfrac{-4x}{x}\cos\dfrac x4 \,\,\boldsymbol\Rightarrow\,\, & -4\cos\left(\dfrac ...
0
votes
1answer
26 views

Solve arc integral

Solve $$\int_{AB}^{}\frac{yds}{\sqrt{x}},$$ where AB - arc $${y}^{2}=\frac{4}{9}{x}^{3}, A(3;2\sqrt{3}), B(8;\frac{32}{3}\sqrt{2})$$ I dont know how to solve this. Can you write a solution?
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vote
2answers
55 views

Surface Integral of a Vector Field Over a Torus

Let $S$ be the surface obtained after rotating $(x-2)^2+z^2=1$ around the $z$-axis. What is the value of $$\int_{S}\mathbf{F\cdot n } dA$$ where $$\mathbf{F}=(x+\sin(yz), y+e^{x+z}, z-x^2\cos(y))$$
2
votes
2answers
203 views

Real Analysis: Prove that $|\int_a^b f(x)dx-(b-a)f(a)|<\frac{1}{2}M(b-a)^2$.

Suppose $|f'(x)|<M$ for all $x\in\mathbb{R}$. Prove that $|\int_a^b f(x)dx-(b-a)f(a)|<\frac{1}{2}M(b-a)^2$. What I have so far as thoughts: I am thinking I will want to use the Mean Value ...
0
votes
1answer
29 views

What is the rule for multiplying in integrals?

What is the rule for finding the integral of the product of two functions? Like this: ∫f(x)g(x)dx
0
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1answer
27 views

thermodynamic related integral question

To calculate the area under region 1 or region 2 or region 1 and 2, my book always divides by 2, and then multiplies by the change in volume.. Say I want area under region 1 and 2, the book does ...
1
vote
0answers
31 views

Integration containing a complex number

Folks, Can I treat the complex number in the following integral: $$\frac1{2\pi}\int\frac1{(1+jw)^2}dw$$ as a constant and move it outside of the integral, like this: ...
1
vote
1answer
53 views

Volume: Disk method

Suppose the curve $y=(x+9)^{1/2}$ from $(0,3)$ to $(7,4)$ is rotated about the axis $x=-5$. Write an integral with respect to $y$ representing the area of the resulting surface of revolution. ...
1
vote
1answer
54 views

I need help to find surface area of the function $\operatorname e^{-3x}$ within interval $[0, 7]$

Can you someone give me a hint to start finding the surface area of the function $f(x) = e^{-3x}$ within interval $[0, 7]$? like which method I should choose ? I tried find area by the following ...
0
votes
2answers
55 views

Evaluate the improper integral

$$\int_0^\infty \dfrac{\arctan(ax)-\arctan(bx)}{x}~\mathrm{d}x$$ where $a$ and $b$ are positive real numbers I could not think of a way where to proceed from. Please help!
0
votes
1answer
49 views

integral partition, real analysis

I'm struggling with this question: If we have $f(x) = x^2 $ and $P_n $ which partitions $[1,3]$ into $n$ sub-intervals, each equal in length,how can I write the formulas for $L(f,P_n)$ and $U(f,P_n)$ ...
0
votes
1answer
36 views

Computing the surface integral of a parabloid

Problem: Solution: I am having difficulty understanding how the author determined the limits of integration of $R$. The author used $\theta=\pi/3\quad to\quad \theta=\pi/2$ and $r=1\quad to\quad ...
3
votes
1answer
95 views

How to prove $\operatorname{Log}(z) = \log(|z|)+i\arg(z)$.

The value of the principal branch of the logarithm can be evaluated by the formula \begin{align*} \operatorname{Log}(z) = \log(|z|)+i\arg(z), \end{align*} where $\arg(z) \in (-\pi,\pi)$ and ...
5
votes
1answer
132 views

Prove $\sin a=\int_{-\infty}^{\infty}\cos(ax^2)\frac{\sinh(2ax)}{\sinh(\pi x)} \operatorname dx$

Derive the integral representation $$\sin a=\int_{-\infty}^{\infty}\cos(ax^2)\frac{\sinh(2ax)}{\sinh(\pi x)}dx$$ for $|a|\le \pi/2$.
0
votes
1answer
31 views

Area in the plane described by inequalities

An area in the plane is specified by the following inequalities: $$x^2 \le y \le \frac6{\sqrt{x}}, \; x\ge 1$$ How do I: draw this area? math this area? Any ideas?
0
votes
1answer
74 views

Find the volume of the solid formed by rotating completely about x-axis the area enclosed by a curve

Question: Find the volume of the solid formed by rotating completely about $x$-axis the area enclosed by a curve. My answer: I drew the curve and the area formed it is between $-2$ and $0$ (the ...
5
votes
2answers
81 views

Inequality associated with Fourier transform

Suppose $$\int_{I_1} x^2|f(x)|^2dx\ge\frac12\int_\Bbb Rx^2|f(x)|^2dx$$ and $$\int_{I_2} x^2|\hat f(x)|^2dx\ge\frac12\int_\Bbb Rx^2|\hat f(x)|^2dx$$ for interval $I_1, I_2$ centered at origin and ...
10
votes
2answers
373 views

Calculate $\displaystyle \int_0^\infty \frac{\ln x}{1 + x^4} \mathrm{d}x$ using residue calculus

I need to evaluate this integral using calculus of residues: $$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$ I know I need to consider $\displaystyle ...
0
votes
0answers
44 views

Expected Value Question (normal Distribution)

I'm trying to calculate $E(X)$ where $f(x)$ is a variable such that; f(x) = 0 , -infinity<=x $$f(x)= \begin{cases} 0 \ , &-\infty \le x \lt c_1\\ x-c_1 \ , & c_1 \le x \lt b \\ b\ , ...
1
vote
1answer
46 views

Find an integral for the area of the surface generated by revolving the curve $y=sin(x)$ between $0 \le x \le \pi$, about the x-axis

So here is my problem: Find an integral for the area of the surface generated by revolving the curve $y=sin(x)$ between $0 \le x \le \pi$, about the x-axis Just thinking about the problem I feel ...
0
votes
3answers
70 views

Decompose integral of derivative and $e^{st}$ (laplace transform)

I'm reading on Laplace transform and stumbled upon the transform of a derived function. Could someone explain me this? $$ \begin{equation} \int_{0^{-}}^\infty \frac{d}{dt}f(t)e^{-st} dt = ...
0
votes
2answers
47 views

Find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ for the curve $x=1+t^2$, $y=t^3-3t$

Is this question simply asking me to find the first and second derivative for the two given equations? I really don't know how to get started with this one and would appreciate any hints. Find ...
5
votes
2answers
72 views

A trigonometic integral with complex technicals

Let $a,b \in \mathbb{R}^+$. Show that $$\int_0^{2\pi}\frac{1}{a^2\cos^2(t)+b^2\sin^2(t)}dt=\frac{2\pi}{ab}$$ Help please! Thanks.
1
vote
1answer
297 views

Area of a square in polar coordinates?

I was attempting, for the exercise of it, to find the area of the a simple square with an infinite number of infinitesimal circle sectors. Let us say this square is $[5 x 5]$. Alas, it's been ...
0
votes
1answer
57 views

If $f$ is a positive integrable function on $(X,\mu)$, why is $\int_0^\infty\mathbb 1_{\{x\in X: f(x)>t\}}(y)dt=\int_0^{f(y)}1dt$?

If $f$ is a positive integrable function on $(X,\mu)$, why is $\int_0^\infty\mathbb 1_{\{x\in X: f(x)>t\}}(y)dt=\int_0^{f(y)}1dt$ ? the exercise was, to show the equality between these 2 ...
1
vote
3answers
63 views

integrate the following equation (what am I doing wrong here 2)

Here is the equation: $$\int 3x \sqrt{1-2x^2}dt$$ Here is my answer: $$ \dfrac14 \int (1-2x^2)^{1/2} . 3x = -\dfrac14 \dfrac{(1-2x^2)^{3/2}}{3/2} = -\dfrac14 \cdot \dfrac23 (1-2x^2)^{3/2} + c$$ ...
0
votes
3answers
107 views

Problem with integrating $\int t^2\sqrt t\, dt$

Here is the equation: $$\int t^2 \sqrt{t}dt$$ Here is my answer: $$\int t^2 \sqrt{t} = \dfrac12 \int t^{1/2}2t = \dfrac12 \dfrac{t^{3/2}}{3/2} = \dfrac12 \cdot \dfrac23 t^{3/2} + c$$ whereas here is ...
0
votes
1answer
45 views

Simplifying integral with Bessel Function

How do I get from Step 1 to Step 2? Do I apply integration by parts?
2
votes
2answers
41 views

integrate the following equation

here is the equation: here is my answer: the correct answer: $-\sqrt {1 - 2x} +c$
0
votes
1answer
79 views

Help with $\int \frac 1{\sqrt{a^2 - x^2}} \mathrm dx$

$\int \dfrac 1{\sqrt{a^2 - x^2}} \,dx$ In the second passage it become $\int \dfrac 1{a\sqrt{1 - \left(\frac xa\right)^2}}\,dx$ So can someone explain me what kind operation is done at ...
4
votes
1answer
91 views

Integral of $x/(e^x-1)$

Problem: Let $f_1(x)=\dfrac{x}{e^x-1}, f_2(x)=\dfrac{x}{e^x+1}$. Show that $f_1,f_2$ are Lebesgue-integrable and $\int_{(0,\infty)}f_1 d\lambda=\sum_{n\in\mathbb{N}}\dfrac{1}{n^2}$ ...
0
votes
0answers
53 views

problem integrating a dirac comb

Let: $$h(t)=\frac{\sin(\pi t(2N+1))}{\sin(\pi t)}$$ $$I=\int_\frac{-1}{2}^\frac{1}{2} h(t) dt$$ when $N\rightarrow\infty$ , obviously (with a change of variable $v=\pi t(2N+1)$ ): ...
-6
votes
2answers
74 views

Finding this integral: $\int_{0}^{2} \sin(x)\cdot\sin(x)$

Find: $$\int_{0}^{2} \sin(x)\cdot\sin(x)$$ I'm trying to solve this integral using the recursive method but it always give me $0=0$. Why I get this?
4
votes
2answers
129 views

Integral $\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx$

I am working on the improper integral: $$\int_0^{\infty}\frac{e^{-x}-e^{-2x}}{x}dx$$ This function does not have an elementary anti-derivative, so here is what I did: define: ...
0
votes
1answer
63 views

calculate double integral on quadrangle domain by changing of variables

Problem: calculate double integral $\iint_D dA$ on quadrangle domain $D$ give by vertices $(1,1),(4,2),(2,4),(3,5)$. How can I change variables to get double integral on form $\int_a^b ...
4
votes
4answers
140 views

How find this integral $\int\frac{1}{1+\sqrt{x}+\sqrt{x+1}}dx$

Question: Find the integral $$I=\int\dfrac{1}{1+\sqrt{x}+\sqrt{x+1}}dx$$ my solution: let $\sqrt{x}+\sqrt{x+1}=t\tag{1}$ then $$t(\sqrt{x+1}-\sqrt{x})=1$$ $$\Longrightarrow ...
1
vote
1answer
98 views

Integrate a function with respect to a different variable?

I'm suppose to integrate this function,$$\int\left(e^{3t}-\frac{3}{t}\right)dx$$ I took the antiderivative and got the incorrect answer, the correct one being: $$\left(e^{3t}-\frac{3}{t}\right)x$$ ...
4
votes
1answer
100 views

Double integral $\iint_D\arctan e^{xy}\,dy\,dx$

Evaluate the double integral: $$\iint_D\arctan e^{xy}\,dy\,dx$$ where $D:\{(x,y)\in\mathbb{R}^2: x^2+y^2\leq 4x\}$ For thie integral , since our function does not have elementary anti-derivetives ...
2
votes
1answer
50 views

How to evaluate the following integral?

I have the integral $$ \int \limits_{0}^{\infty} \frac{dx}{(x + 1)\sqrt{(x + 1)^{3}a^2 + (1 - a^2)}}. $$ How to evaluate it? I tried to reduce it to beta-function, but I failed.
0
votes
1answer
68 views

Fredholm integral equation of first kind

I want to solve the Fredholm integral equation of first kind: $$ \int_L K(x,y)U(y)dy = f(x) $$ in these equation the function $U(y)$ is the unknown and the so-called kernel $K$ and the right hand side ...
0
votes
3answers
117 views

Where does $\frac12$ come from in this integration?

I have the following integration that was given as an example but I do not understand where $\color{red}{\frac12}$ comes from. Please explain. $$\int (2x-1)^4 ...
4
votes
3answers
227 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
1
vote
1answer
18 views

Integrating strips normal to the curve

How do you go about finding the following area under the curve where strips are supposed to be normal to the curve?
1
vote
1answer
100 views

Evaluating $I(a,n) = \int^{\infty}_{-\infty}{e^{iax - nx^2}}\, dx$ for real $a$ and real $n > 0$ with Jordan's Lemma

Given that $e^{-nx^2}$ does not have any singularities, I believe $I(a,n) = 0$. Is this a correct application of Jordan's Lemma? $$I(a,n) = \int^{\infty}_{-\infty}{e^{iax - nx^2}}\, dx$$
1
vote
1answer
317 views

Pumping Water out of Parabolic tank?

First of all, I understand how to do the integration part of this problem, but I am confused about the setup. Here is the question: Use integration to find the work done pumping all the water ...