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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1answer
16 views

Integral, partial fractions, need explanation for how to get from one step to another.

Can someone explain how they go from the red step to the blue one?
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1answer
24 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
1
vote
1answer
22 views

Proving a reduction formula. $\cos^n (2x)$

Establish a reduction formula for $$\int \cos^n (2x)dx$$ My attempt, Let $I_n=\int \cos^n 2x dx$ $=\int \cos^{n-1}2x (\cos 2x dx)$ Let$$u=\cos^{n-1}2x$$ $$du=-2(n-1)\cos^{n-2}2x (\sin 2x)dx$$ ...
0
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0answers
19 views

Bounding an integral

I'm trying to show that the following integral ( a solution for the non-homogeneous transport equation ) has this bound: $$ \begin{equation*} \left\|{ \int_{0}^{t} f(x+b(w-t),w) dw ...
1
vote
1answer
28 views

Sign of the error in Simpson's rule

Let $f : [a,b] \to \mathbb{R}$ be a $C^\infty$ function. The Riemann integral $I = \int_a^b f(x)\,dx$ can be approximated by using Simpson's rule: $$I \approx S = \frac{b-a}{6} \left[ f(a) + 4 ...
3
votes
0answers
30 views

An alternative way to determine when $\int_{0}^{\infty} \cos(\alpha x) \prod_{m=1}^{n} J_{0}(\beta_{m} x) \, dx =0$

Let $J_{0}(z)$ be the Bessel function of the first kind of order zero, and assume that $\alpha$ and $\beta_{m}$ are positive real parameters. When $|z|$ is large in magnitude and $-\pi < \arg(z) ...
1
vote
1answer
58 views

Integrate $\int{ \left( \frac{1-x}{1+x} \right)^\frac{3}{2}dx}$

Integrate $$\int{ \left(\frac{1-x}{1+x} \right)^\frac{3}{2}dx}$$ I guess that there is sub $x = \cos t$ so integral gets to $$\int{ \left(\tan \frac{t}{2} \right)^3 d\cos t}$$ then I used that $\sin t ...
0
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1answer
24 views

Surface are of a curve $y=\sin \left(\frac{\pi x}{6} \right)$ rotated about the $x$ axis.

I'm doing a problem involving finding the surface area of the curve for $y=\sin \left(\frac{\pi x}{6} \right)$, rotated about the $x$ axis, for $[0 < x < 6]$. I got as far as $\frac{72}{\pi} ...
0
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1answer
21 views

Problem with convolution, insecure

$$f(t)= t^2\cdot u(t),\quad g(t)=t^4\cdot u(t)$$ I know that I need to use convolution theorem to solve this problem, but I really don't know what to do with step functions. Do I need to include ...
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0answers
25 views

Evaluating this surface integral

I want to evaluate the following surface integral $$\int_S d S \, \, \exp\left(\frac{\Gamma^2 (x^2 \sigma_x^2 + y^2 \sigma_y^2 + z^2 \sigma_z^2) - 2 c R \Gamma (x x_0 + y y_0 + z z_0)}{2 c^2 ...
3
votes
4answers
70 views

Clever way of calculating the integral $ \int \frac{dt}{t^2\sqrt{t-2} } $

$$ \int \frac{\text{d}t}{t^2\sqrt{t-2} } $$ I know it can be calculated using somewhat complicated substitutions, but is there possibly some clever way of solving that type of integral? I don't ...
1
vote
0answers
13 views

Integral of least squares and general rules of integration to solve the integral.

My calculus is very rusty and I am interested to know if the following is solvable: $$ \int_0^{\pi}( \log( \frac {(x_0 + e^{-i\omega})(x_0 + e^{i\omega})(x_1 + ...
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2answers
53 views

How would you calculate $(200\int_0^\infty e^{-0.8t}-e^{-1.8t}\,dt)/(250\int_0^\infty e^{-0.8t} \,dt)$?

$$\frac{200\int_0^\infty e^{-0.8t}-e^{-1.8t} \, dt}{250\int_0^\infty e^{-0.8t} \, dt}$$ I am confused as to how you would integrate the e's from zero to infinity. What steps would you take? By the ...
1
vote
0answers
29 views

reduction formula for $\int \tan^n (2x)dx$

Establish a reduction formula for $$\int \tan^n (2x)dx$$ My attempt, Let $I_{n}=\int \tan^n (2x)dx$ $=\int \tan^2 (2x) \tan^{n-2} (2x)dx$ $=\int (\sec^2 (2x)-1)\tan^{n-2}(2x)dx$ $=\int ...
1
vote
0answers
11 views

Prove that the condition $x(\tau)>\xi$ of a divergent integral implies that $x(t)>\xi$

Let $ E, J \subset \mathbb R$ be open intervals and let functions $h:J \to \mathbb R$ and $g: E \to \mathbb R$ be continuous. let $\xi \in E$ and assume that $g(\xi)=0$. Define $f:J \times E \to ...
2
votes
3answers
63 views

Calculate $\int_0^1 \ \int_0^1 \ x \sin \lvert x^2-y^2 \lvert \; dx \; dy$

$$\int_0^1 \ \int_0^1 \ x \ \sin \lvert x^2-y^2 \lvert dx \ dy $$ $$\int_0^1 \frac{1}{2} \Big[ \sin \lvert x^2-y^2 \lvert \Big]_0^1 \ dy= \int_0^1 \frac{1}{2} \Big( \sin \lvert 1-y^2 \lvert - ...
4
votes
6answers
117 views

integrate $\int \frac{dx}{x\sqrt{1-x}}$

$$\int \frac{dx}{x\sqrt{1-x}}$$ $$\int \frac{dx}{x\sqrt{1-x}}$$ $u=1-x$ $du=-dx$ $$-\int \frac{du}{(1-u)\sqrt{u}}$$ $a(1-u)+b\sqrt{u}=1\Rightarrow a-au+b\sqrt{u}=1$ $a=1\Rightarrow ...
2
votes
2answers
53 views

How to solve without involving hyperbolic function.

How to solve this integral without involving hyperbolic functions? $$\int \frac{1}{4-5\sin^2 x}dx$$ The answer is $\frac{1}{4}(\ln (\sin x+2 \cos x)-\ln(2\cos x-\sin x))+c$
2
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1answer
36 views

Imaginary number and absolute value integral - Fourier transform

I came across this integral problem: $$\hat f(\xi)=\int_{-\infty}^{+\infty} e^{-|x|+xi\xi}dx$$ Now I know how to integrate simple absolute value functions like: $\int_{-2}^{4}|x-2| dx$, we just ...
1
vote
1answer
31 views

Density of $L^\infty(\Omega)h$ in $L^p(\Omega)$ where $h \in L^p(\Omega)$

Let $(\Omega,\mu)$ be a finite measure space. Suppose $1\leq p <\infty$. Let $h$ be an element of $L^p(\Omega)$ with $h >0$ a.e.. How show that the subspace $L^\infty(\Omega)h=\{ f h\ :\ f\in ...
0
votes
2answers
51 views

Indefinite trignometric integral

I tried $u$-substitution and $uv$-substitution, can't seem to figure this out... any help would be appreciated! Question: $$\int\frac{x}{\cos(x)}\,dx$$ Thanks!!!
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3answers
27 views

How to find differentiation and integration of curves in general?

Graph of function $f(x)$ How do I go about finding integration and differentiation of curves like these which yield other curves?
0
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1answer
44 views

Finding Value of C to Maximize Area

f(x)=$xe^{-\sqrt x}$ Find the value of c, such that the area bounded between the graph, the x-axis, x=c, and x=c+1 is maximized. Find the maximum area. I don't know where to start with this one. I ...
0
votes
3answers
26 views

How to account for solids of revolution around vertical lines to the right of the x axis?

I'm trying to find the volume of a solid created by rotating the region enclosed between $x=y^2$ and $x=1$ around the line $x=8$. Noting that the intersections of the functions occur at $(0,0)$ and ...
0
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1answer
15 views

Vector integral in $N$ dimensions.

In $N$ dimensions I want to do an integral of the flux through an $N-1$ dimensional surface. The usual vector calculus integration theorems help by allowing integration around the perimeter of the ...
0
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1answer
23 views

Trouble getting between steps when solving integral

I've having a lot of trouble trying to figure out how they're getting from the step in blue to the one in red. Can some one please explain that?
0
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0answers
18 views

Estimate integration

Suppose $n<<k$ and $n<i<k$. How can I estimate this integral: $\int_{1}^{k-i} x^{-0.4}*(x+i-1)^{-0.4}dx$. I would like to get the result in the form of O(f(k,i)). Since the integrand is ...
2
votes
1answer
34 views

Finding Fourier series constant and integral

I have been studying Griffith's Intro to Electrodynamics. I am studying differential equations and Fourier series. I am studying the problem discussed here: Why is this allowed? ("Fourier's ...
2
votes
1answer
21 views

Volume of region in the first octant bounded by coordinate planes and a parabolic cylinder?

Find the volume of the solid region in the first octant bounded by the coordinate planes, the plane $y + z = 2$ and the parabolic cylinder $x = 4 - y^2$. I have a final answer, I would just like to ...
1
vote
2answers
68 views

Integrate $\int \frac{x^7}{(1-x^4)^2}dx$

$$\int \frac{x^7}{(1-x^4)^2}dx$$ I have tried to simplify the expression, to use U substation, any idea where to start from?
1
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0answers
31 views

Is there a close form expression for the integral $ \int_a^b |x-c|^n e^{-x^2/2} $

Is there a close form expression for the integral \begin{align} \int_a^b |x-c|^n e^{-x^2/2} dx \end{align} by close form I mean it can be in terms of well know functions such as $Q$-function, ...
0
votes
2answers
64 views

How do I evaluate a series? [on hold]

In this specific example, I don't understand the steps of evaluating this series: \begin{align} &\frac{12}{n}\left(\left[\sum_{i=1}^n-7\right]+\sum_{i=1}^n\left[\frac{-12}{n}\cdot ...
1
vote
2answers
47 views

Arc length of Archimedes Spiral $ r = \theta $ from $ 0 \le \theta \le 2\pi$

The equation of the Archimedes spiral is given by $$r = \theta$$ The formula for calculating the Arc Length is given by $$L = \int^b_a\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$ The ...
2
votes
0answers
38 views

Asymptotic behaviour of an integral depending on a parameter

I am trying to compute the asymptotics on $t$ of the following integral: \begin{equation} I(t)=\int_{\mathbb{R}^{n}}e^{-|\lambda|^{2}/2t}\prod_{i<j}\left( e^{\lambda_{j}/t}-e^{\lambda_{i}/t} ...
1
vote
7answers
100 views

Integrate $\int \frac{x\cos x}{\sin^2x}dx$

$$\int \frac{x\cos x}{\sin^2x}dx$$ $$\int \frac{x\cos x}{\sin^2x}dx=\int \frac{x\cos x}{1-\cos^2x}dx=\int \frac{x\cos x}{(1-\cos x)(1+\cos x)}dx$$ How can I find the two fractions? if there are ...
3
votes
0answers
49 views

How to calculate the area of the visible parts of a 3D PieChart?

I have created a 3D Pie Chart whose major feat (among the others) is to be rotated: I did it to demonstrate how the visual perception of data in a Pie Chart can be distorted depending on the ...
-1
votes
1answer
17 views

how to prove if $f$ is integrable and $f'$ too then the limit of $f$ is zero when $x$ go to infinity? [on hold]

If $f$ is a real function on $\mathbb R$ and we have $\int_1^\infty |f(x)|dx < \infty$ and $\int_1^\infty |f'(x)|dx < \infty$ then $\lim_{x\to\infty}f(x)=0$ ?
1
vote
0answers
42 views

Compute $\frac{d}{dt}\int_0^t e^{x(s)}ds$, where $x$ is a standard Brownian motion.

How to compute the following differentiation? Is there a general rule that can be applied? $$\frac{d}{dt}\int_0^t e^{x(s)}ds$$ in the case of $x=W$ where $W$ is a standard brownian motion, is there ...
-2
votes
2answers
89 views

Evaluate $\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$ [on hold]

As a consequence of this Q, I need some help evaluating the following integral: $$\int_{-\infty}^{\infty}x^2 e^{-\alpha x^2+\beta x}dx$$ Integration by parts wouldn't simplify things and I guess that ...
0
votes
1answer
36 views

finding area using iterated integral

I am trying to find the area enclosed between $f(x)=\sin x$ and $g(x)=\cos x$ between $x= \pi/4$ to $x = 5 \pi/4$. I got $\int_{\pi/4}^{5\pi/5}\int_{\cos x}^{ \sin x} dydx$. But I am not getting the ...
4
votes
1answer
39 views

How would you integrate this?

If we had the following integral: $$\int_{a}^{b} {\big(1+x^2 \big)^s} \space dx$$ Where $s$ is not given. Is there any general formula for this integration that works for all $s\in \mathbb{R}$?
1
vote
3answers
27 views

Proof of integral involving the inverse hyperbolic secant and cosent

We know that $$ \int \frac{dx}{x \sqrt{a^2 \pm x^2} } = -\frac{1}{a} \ln \frac{a+ \sqrt{a^2 \pm x^2}}{\lvert x\rvert }+C$$ I tried proving this integral setting $x = a \ \mathrm{csch} \ u $ and using ...
2
votes
2answers
37 views

Advanced calculus, Riemann integral.

If $f$ is (Riemann) integrable on $[a,b]$ and if $\int_{a}^{b} fh=0$ for all continuous function $h$, then $f(x)=0$ for all points of continuity of $f$. I know if we have $f$ being continuous on ...
5
votes
1answer
101 views

Improper integral: $\int_1^\infty\frac{\sin(\sqrt{x})}{\sqrt{x}}dx $.

mathematica is reporting that the improper integral $\int_1^\infty\frac{\sin(\sqrt{x})}{\sqrt{x}}dx $ coverges to $2\cos(1)$. However, when I try to confirm this by actually integrating it using ...
0
votes
1answer
26 views

Proof of integral involving hyperbolic tangent

We know that $$ \int \frac{dx}{a^2-x^2} = \frac{1}{2a}\ln \left| \frac{a+x}{a-x}\right| +C$$ (That absolute value sign is supposed to be longer. I apologize for ignorance on how to make that longer on ...
1
vote
0answers
36 views

Flux Through a Closed Curve - Orientation

I want to compute $$\int_{C}\boldsymbol{F}\cdot\boldsymbol{n}\, ds\qquad\quad \boldsymbol{F}=\langle x, y^2\rangle$$ where $C$ is the curve given by the triangle with vertices $(-1,0)$, $(0,1)$ and ...
2
votes
0answers
18 views

Changing integration bounds

I came accross this line: $$x\in [0,1],y\in [0,1]$$ $$E(Y|x)=\int_y y*g(y|x)dx=\int _0^1y*g(y|x)dy$$ Can someone please explain how the second equality holds! Thanks.
3
votes
1answer
31 views

Trigonometric integrals and limits

Show $$\lim_{N\to\infty}g_N(\theta_N)=2\int^\pi_0\frac{\sin x}{x}dx-\pi,$$ where $$g_N(\theta_N)=\int_0^{\theta_N}\frac{\sin[(N+1/2)x]}{\sin(x/2)}dx-\pi,$$ $$\theta_N=\frac{\pi}{N+1/2},$$ and ...
3
votes
2answers
53 views

What is the the integral of $\sqrt{x^a + b}$?

How do you evaluate $\displaystyle\int\sqrt{x^a + b}\,\,\text{dx}$, where $a \neq 0$ and $a \neq 1$? For example, how do you evaluate $\displaystyle\int\sqrt{x^2 + 1}\,\text{dx}$? If we let ...
4
votes
0answers
71 views

Integral involving power of trigonometric functions

I'm having a technical problem evaluating the following integral: $$\int_{r=0}^1\int_{\theta=0}^{\pi \over2} \cos^{2\epsilon -1}\theta \sin^{\epsilon-1}\theta e^{-ikr\sin^\epsilon\theta}d\theta dr$$ ...