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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1
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0answers
14 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
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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
48 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
5answers
109 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
46 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
votes
0answers
26 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
27 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
1answer
41 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!!!
0
votes
3answers
24 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
votes
1answer
32 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
23 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
votes
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
22 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
votes
0answers
16 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
33 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
19 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
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2answers
66 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
29 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
61 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
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2answers
46 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
36 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
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7answers
96 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
45 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
40 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
78 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
35 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
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3answers
24 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
91 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
31 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
17 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
30 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
51 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
56 views

Integral involving power of trigonometric functions

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

Regarding Apostol's theory of integration

I have some questions regarding the theory of integration as discussed in Tom Apostol's Calculus. Integration is defined using step functions. My question is, is this definition he presents equivalent ...
1
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3answers
60 views

integrate $\int \sin^{4}x\cos^{2}x$ [duplicate]

$$\int \sin^{4}x\cos^{2}xdx$$ $$\int \sin^{4}x\cos^{2}xdx=\int (\sin x \cos x)^{2}\sin^2xdx=\int \left(\frac{\sin^{2}2x}{2}\right)\left(\frac{1}{2}-\frac{\cos2x}{2}\right)dx=\int ...
1
vote
6answers
107 views

Integrate $\int_{-\infty}^\infty xe^{-\alpha x^2+\beta x}dx$ [duplicate]

I am familiar with the gauusian integral $$\int_{-\infty}^\infty e^{-\alpha x^2+\beta x}dx=\sqrt{\frac{\pi}{\alpha}}e^{\beta^2/(4\alpha)}$$ Could anyone help me to find out the value of the following? ...
0
votes
1answer
41 views

Limit with number of integrals tending to infinity

Let $F_0(x) = \ln x$. For $n \geq 0$ and $x >0$, let $F_{n+1}(x)=\int_0^x F_n(t)dt$. Evaluate $$\lim_{n \to \infty} \frac{n! F_n(1)}{\ln n}$$ Because the final intergal is from $0$ to $1$, I ...
1
vote
1answer
47 views

To prove or refute: $\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} f\left( \frac{n}{N} \right) = 1$ then $f \in R\left( \left[ 0, 1 \right] \right)$ [on hold]

Let $f : \left[ 0, 1 \right] \to \mathbb{R}$ such that $$\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} f\left( \frac{n}{N} \right) = 1.$$ Then, $f \in R\left( \left[ 0, 1 \right] \right)$ and ...
2
votes
1answer
55 views

Integral of $\int_0^{2\pi}{e^{({x\sin\theta)}}\ \text{d}\theta}$

I would like to compute $$f(x)= \int_0^{2\pi}{e^{({x\sin\theta)}}\ \text{d}\theta}$$ I already know that it equals $2\pi\sum_{j\geq0}{(\frac{x^j}{2^jj!})^2}$. I'm already happy since it provides an ...
-2
votes
1answer
29 views

Evaluate a double integral over a region $R$

Let $R$ be the refion enclosed by $x^2+4y^2\ge 1$ and $x^2+y^2\le 1$. Calculate $$\iint_R \lvert xy\rvert\,dxdy$$ I think the answer is $0$ because the area in the positive quadrants cuts the ...
2
votes
3answers
69 views

Dealing with integrals of the form $\int{e^x(f(x)+f'(x))}dx$

Integrals of the form $$\int{e^x(f(x)+f'(x))}dx$$ are very common. And I have seen this form appearing in several exam papers.But the problem I face with this particular type of integral is finding ...
3
votes
1answer
93 views

integrate $\int \frac{dx}{4+3sin2x}$

$$\int \frac{dx}{4+3\sin (2x)}$$ $u=2x$ $du=2dx$ $$\frac{1}{2}\int \frac{du}{4+3\sin(u)}$$ $v=\tan(\frac{u}{2})$ $du=\frac{2dv}{1+v^2}$ \begin{align*} \frac{1}{2}\int ...
0
votes
0answers
25 views

Determination of a formula of an area of shape limited by curves using integrals?

I did two examples of determining an area limited by curves and I'm not sure if I did them right. I would really appreciate if someone checked my solution and fixed any possible mistakes. Ex.1 Area ...
1
vote
0answers
25 views

Discontinuous parametric integral function

Is there an example of a function $f:[0,1] \times [0,1] \to \mathbb{R}$ such that for all $x \in [0,1]$ the function $\phi(y) = f(x,y)$ is continuous in $y$ and for all $y \in [0,1]$ the function ...
1
vote
1answer
29 views

Calculation of area in 2 definite integrals given function $y=x^2$

Here is a graph for $y=x^2$ Given that the area in blue is equal to the area in pink, find a in terms of b and solve for a. My attempt: From the graph I can see that :$a^2=b$ and $a=\sqrt b$ ...
0
votes
1answer
19 views

Finding Intervals after Changing the Order of Integration

Problem: Let $f(x) = \int_0^x e^{t^2} \,dt.$ Find the average value of f on the interval $[0, 1]$. Thoughts: $\int_0^x e^{t^2} \,dt$ is a non-elementary integral. The average value of f, I believe, ...