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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0
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2answers
44 views

Is the quotient of two continuous functions absolutely integrable?

(True or False) Suppose that $h$ is absolutely integrable on $(a,b)$. If $f$ is continuous on $(a,b)$, if $g$ is continuous and never 0 on $[a,b]$, and if $|f(x)|\leq{h(x)}$ for all $x\in[a,b]$, then $...
0
votes
1answer
83 views

Find the volume of a body bounded by $z = x^2 + 3y^2, z= 8-x^2-5y^2$.

Find the volume of the body bounded by the elliptic paraboloids $z = x^2 + 3y^2, z= 8-x^2-5y^2$. What i did? First of all i intersected two functions and got the desire area is a ellipse $x^2+4y^...
3
votes
0answers
138 views

Is $h(\mathbf{a},b)=\int_\Omega f(\mathbf{x})e^{-g(\mathbf{x})}\mathrm{d}\mathbf{x}$ differentiable?

Let $f\colon\Bbb{R}^n\to\Bbb{R}$ be an affine function and $g\colon\Bbb{R}^n\to\Bbb{R}$ be a non-negative function. We define $h\colon\Bbb{R}^n\times\Bbb{R}\to\Bbb{R}$ as follows $$ h(\mathbf{a},b)=\...
1
vote
0answers
29 views

Compare $\int_{-\infty}^{+\infty} f(x) dx \quad\text{ and }\quad \lim_{t\rightarrow +\infty} \int_{-t}^{t} f(x) dx$

I would like to compare these two integrals $$\int_{-\infty}^{+\infty} f(x) dx \quad\text{ and }\quad \lim_{t\rightarrow +\infty} \int_{-t}^{t} f(x) dx$$ My Thoughts: Let $\displaystyle I=\...
2
votes
4answers
82 views

Integral how to choose the good substitution?

I have a problem with this integral. I don't know what kind of substitution to use. $$\int t\sqrt{1+9t^4}dt$$
1
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1answer
53 views

How do I evaluate the following double integral over a general region

How do I evaluate the following integral $$ \int_{-1}^1 \int_{\arccos(y)}^{\pi} \sin(x)\sqrt {1+\sin^2 x} \,dx\,dy $$ I attempted to change the limits for the integrals $$ \int_0^\pi \int_{\cos(x)}^...
0
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1answer
83 views

What is the primitive of $f(x) = (x^a + b)^\frac{1}{a}$?

In studying a physical problem I was stopped by an integral that can be written in this clean way $$ \int_0^K (x^a + b)^\frac{1}{a} dx \qquad a,b \in \mathbb{R} \qquad x,K \in \mathbb{R}^+ $$ I tried ...
2
votes
4answers
111 views

what's $\int \sin (e^x)\ dx$ ??

I was working on a physics problem and I faced this integral: $$\int \sin(e^x) \, dx =$$ I tried to solve it but I could not.
0
votes
0answers
14 views

Replicate Matlab integrator block in MS excel

I created a simple diagram to solve ordinary differential equation as shown below. Simple ODE I was trying to compute the result of xf_dot manually in Ms Excel but I did not get the same answer ...
0
votes
0answers
47 views

Definite integral of product of Algebraic, Bessel's function, and exponential involving algebraic function from limits $x = u$ to $x = \infty$

I'm deriving Cumulative Distribution function (CDF) of a random variable $\Gamma$ as given below: \begin{equation} \Gamma = \frac{XY}{aX+bY+cZ+d} \end{equation} After lots of manipulation I'm stuck ...
1
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2answers
44 views

solution of an improper integral.

I was solving following improper integral: $$ \int\limits_0^\frac{\pi}{2}\frac{log~x}{x^a}dx $$ where $a<1$. My attempt: $0$ is the only point of discontinuity. So, $\frac{log~x}{x^a}\leq \...
4
votes
2answers
86 views

Computing the Integral $\int r^2 \text{J}_0(\alpha r) \text{I}_1(\beta r)\text{d}r$

I encountered the following integral in a physical problem $$I=\int r^2 \text{J}_0(\alpha r) \text{I}_1(\beta r)\text{d}r$$ where $\text{J}_0$ is the Bessel function of first kind of order $0$ and ...
2
votes
1answer
66 views

How to solve $y' + y^2 - 2y\sin x + \sin^2x = \cos x$

How to solve the following equation? $$y' + y^2 - 2y\sin x + \sin^2x = \cos x$$ It is necessary to determine the type and total solution. Help me please.
0
votes
1answer
21 views

Fourier transform , of a function computation

I need help with this Fourier transform computation. $$F(w)=\int_{-\infty}^{\infty} e^{-|x|+ix}e^{-iwx} \, dx$$ Need help to compute.
6
votes
0answers
127 views

Calculation of $\int_{0}^1 \frac{\sin(\ln^4(1-x))}{x}~dx$

$$I=\int_0^1 \frac{\sin(\ln^4 (1-x))}{x}dx$$ What is the closed-form evaluation of this integral? I honestly do not have a single clue how to solve this. (There is no application, but it is out of ...
1
vote
2answers
58 views

If $f$ is positive, decreasing, and continuous on $[0,+\infty)$ show that $\int_0^\infty f(t)dt$ converges iff $\sum_{n=0}^\infty f(n)$ converges

My Work: Integrals take infinitely small steps to get from one term to the next, whereas in a series, the distance between each term must be some tangible value ($|x_n-x_{n-1}|\ge \epsilon$). Thus, ...
1
vote
1answer
35 views

Solve for a variable that is within an integral and the upper bound of the integral

So I took a Calculas test today and this is a problem that I know I did not get correct. I've been thinking for some time on it and this is as far as I got, any corrections or an answer would be great,...
0
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0answers
37 views

Fourier transform

I am having a difficulty understanding how to approach this problem. I have state vector $$f(x)=e^{-|x|+ix}$$ and observable $$P=-id\dfrac dx $$ I want to find fourier transform of $f$? and the ...
1
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1answer
47 views

Fourier transforms having compact support

As we know, the fourier transform is a map $\mathcal{F}:L^1\rightarrow C_0$ (all with domain $\mathbb{R}$). Can one characterize the space of $f\in L^1$ such that $\mathcal{F}$ has compact support, i....
2
votes
1answer
37 views

Link between fundamental theorem of calculus and integral with parameters

My problem is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. We know that $$F:x\mapsto\int_0^xf(t)dt$$ is the primitive of $f$ that vanishes in $0.$ But i can rewrite $F$ ...
0
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3answers
64 views

How to evaluate integral with radian limits?

$$\int_{\pi/3}^{\pi/6} \csc \theta \cot \theta\, d\theta $$
0
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2answers
61 views

What is the interpretation of the fundamental theorem of line integrals?

The fundamental theorem of line integrals is: $$\int_{a}^{b} \nabla F \cdot \vec{dr} = F(r(b)) - F(r(a))$$ for some curve traced by $r$. What is the intuition for why this is true? The proof is ...
0
votes
1answer
40 views

Finding an Integral Value given other Integral Values?

I am studying for an upcoming Calculus exam, and was hoping someone here could explain how to do the following. \begin{align} \int_1^9 f(x)\,\mathrm dx&=-1\\ \int_7^9 f(x)\,\mathrm dx&=5\\ \...
0
votes
1answer
61 views

Evaluate $\int^e_1\frac{xe^x+1}{x}f(e^x+\ln x)\,dx$, where $g'(x)=f(x)$.

Suppose $g'(x)=f(x)$ for all $x$. Evaluate $$\int^e_1\frac{xe^x+1}{x}f(e^x+\ln x)\,dx$$ Also, what are these type of questions called?
0
votes
1answer
18 views

kinetic energy interation

I was reading the following article http://www.askamathematician.com/2015/03/q-why-does-kinetic-energy-increase-as-velocity-squared/ I don't understand the math of the explanation at the final of the ...
1
vote
1answer
74 views

Integration $\int_0^{2\pi} \frac{\cos^2 3\theta d\theta}{5-4\cos2\theta}$ by using residues

$$\int_0^{2\pi} \frac{\cos^2 3\theta d\theta}{5-4\cos2\theta}$$ By substituting $\cos m\theta$ to $\frac{z^m+z^{-m}}{2}$ and $d\theta$ to $\frac{-i}{z}dz$,I get $$\int_0^{2\pi} \frac{\cos^2 3\theta ...
0
votes
1answer
47 views

Uncertainity regarding the proof from Rosenlicht's “Introduction to Analysis”

I would be very grateful if somebody could explain me the part in the red rectangle in the proof below. I don't quite get it why the author claims that if $\delta$ is sufficiently small then $p+1\le q-...
0
votes
3answers
89 views

why $\int \sqrt{(\sin x)^2}\, \mathrm{d}x = \int |\sin x| \,\mathrm{d}x$

May be this is a stupid question but why $$\int \sqrt{(\sin x)^2} \,\mathrm{d}x = \int |\sin x| \,\mathrm{d}x$$ instead of $$\pm \int \sin x \,\mathrm{d}x$$ I think may be because it violates the ...
0
votes
0answers
59 views

How to get asymptotic form of the integrals with special functions?

I got difficulty when I try to plot I(x) for $m=1$ and $t=0.2$. The questions is how to get the asymptotic form of the following integral? $I(x,t)=\int_{0}^{\infty} \frac{f(y)}{2 \sqrt{\pi t}} \...
-1
votes
1answer
94 views

Volume bounded by a half-cone and a spherical surface [closed]

I'm having trouble computing the integral below. If someone could explain, with precise details, the method for obtaining an answer so I can successfully solve similar problems in the future, it would ...
1
vote
2answers
36 views

Integration of $|e^{-(2+j)t}|^2$

The integration of $|e^{-(2+j)t}|^2$ from zero to infinity is $1/4$ when I separate above as $|e^{-2t}|^2 \cdot |e^{-2jt}|^2$ and integrate. $|e^{-2jt}|$ was taken as $1$. But when I integrate the ...
3
votes
0answers
35 views

Green's Theorem with respect to a given polar region.

Using Green's Theorem, compute the counterclockwise circulation $I$ of $\vec{F}=\langle-\sqrt{x^2+y^2},\sqrt{x^2+y^2}\rangle$ around the region defined by the polar coordinate inequalities $7 \leq ...
0
votes
1answer
79 views

Calculate $\int_{\gamma}\frac{1}{z}dz$ by definition.

I'd like to calculate $$\int_{\gamma}\frac{1}{z}dz$$ where $\gamma$ is the contour of a circle that doesn't contain the origin. I'd like to do it using the definition of integral along a curve, ...
1
vote
1answer
59 views

Integration over complex plane

I have a problem with the following integral $$\int_{-\infty}^{\infty}\frac {x\sin x}{x^4+1}$$ Can someone please help me with the way the solution goes? I would highly appreciate it Thanks in ...
3
votes
1answer
53 views

Find $ \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$.

Using Residue Theorem find $\displaystyle \int_0^{2\pi} \frac{1}{a^2\cos^2 t+b^2 \sin^2 t} dt \;; a,b>0$. My Try: So, I am going to use the ellipse $\Gamma = \{a\cos t+i b \sin t: 0\leq t\leq 2\...
2
votes
1answer
26 views

How to compute the Laurent series

How would you find the Laurent Series for this function: $$f(z) = z^2 + \frac {1}{z^2 - 2z + 10}$$ So I know the singular points are $ z = 1+3i$ and $z=1-3i$ and then I changed the function into $$...
1
vote
0answers
84 views

Vector field flux calculation

Consider the vector field: $$F=((y+1)e^{y^2}\sin(y^3),0)$$ and the curve consisting of the three line segments $A$, $B$ and $C$. Where $A$ goes between $(1,-1)$ and $(1,1)$. $B$ goes between $(1,1)...
5
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1answer
196 views

Verify integration of $ \int\frac{\sqrt{2-x-x^2}}{x^2}dx $

This is exercise 6.25.40 from Tom Apostol's Calculus I. I would like to ask someone to verify my solution, the result I got differs from the one provided in the book. Evaluate the following integral: ...
0
votes
1answer
66 views

Prove that if two continuous functions have equal integrals over interval then the functions are equal at a point c in interval.

Here is the statement I am struggling to prove. Suppose that $f(x)$ and $g(x)$ are continuous functions on [a,b], and that $\int^b_a f(x)=\int^b _a g(x)$. Prove there exists a point $c \in [a,b]$ ...
0
votes
2answers
69 views

Integral of $(x^2 \sin (x))/(x^4 + 2 x ^2 - 1)$

I recently had this integral on a test $$\int ^\pi _{-\pi} \frac{x^2 \sin (x)}{x^4 -2x^2 -1} dx$$ Mr professor claims this is actually equal to zero because the function is periodic and the top is ...
0
votes
0answers
78 views

Triple integrals - spherical coordinates

Integrate the function: $$f(x,y,z)=1/\sqrt{(x^2+y^2)(x^2+y^2+z^2)}$$ over the region $R$ which is the set of all points outside the sphere $x^2+y^2+(z-1)^2=1$ but inside the sphere $x^2+y^2+z^2=4$. ...
0
votes
1answer
52 views

How to compute this integral

How to compute this integral? $$\int \limits_{C}^{}\frac{z^{a}\,dz}{(z^3+1)^2}$$ Given that $a>0$ and $ C:= \{z:|z+1|=\frac{1}{2}\}$ so I know that $log(-1) = \pi i $ and $z^a := e^{alog(z)}$
1
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0answers
30 views

How to compute this integral on contour

How to compute this following integral? $$\int \limits_{C}^{}\frac{e^{az^2}\,dz}{z^4+1}$$ Given $ a>0$ and $C:=\{z: |z+1| = 1\}$ is positively oriented. Where should I start? What would $f(z)$...
0
votes
1answer
61 views

how to compute integral

How to compute this following integral? $$\int \limits_{-\infty}^{+\infty}\frac{x\sin(ax)\,dx}{(x^2+4x+13)}, a>0$$ So I started finding singularity points $$-2-3i$$ and $$-2+3i$$ After that I'm ...
0
votes
1answer
17 views

Volume With Cylindrical Coordinates

Find the volume in the first octant bounded by the parabolic cylinder, bounded by $x-1 = y^2$, the circular cylinder $4z + 4y -z^2 - y^2 = 7$ and the plane $x = 0$.. My attempt: I first took the ...
3
votes
3answers
43 views

Find the double integral by changing to polar coordinates [duplicate]

So I have the following double integral $$\int_{-2}^{0} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \sqrt{x^2+y^2} dydx$$ If I integrate with respect to y first I get: $\frac{1}{2}(y\sqrt{x^2+y^...
1
vote
6answers
98 views

if $\int_0^4 f(x)dx = 5$ then $\int_0^2 f(x)dx=2.5$?

If $\int_0^4 f(x)dx = 5$, then is it true that $\int_0^2 f(x)dx=2.5$? I think this is only true if $f(x)$ is constant but not for polynomial function. However I don't know how to explain it in words.
-2
votes
3answers
52 views

Let $Q=[0,\pi]\times[0,\pi]$ , then how to evaluate $\int\int_Q |\cos(x+y)|dxdy$ ? [closed]

Let $Q=[0,\pi]\times[0,\pi]$ , then how to evaluate $\int\int_Q |\cos(x+y)|dxdy$ ? Please help . Thanks in advance
0
votes
1answer
56 views

Software where I can numerically evaluate multivariable integrals over a region?

I need, for example, to evaluate: $$\iiint(x-1)\,dx\,dy\,dz$$ over the region: $$y=0,\, z=0,\, y+z=5,\, z=4-x^2$$ but I have no ways to verify if I did it right. I needed a form of numerical ...
0
votes
1answer
48 views

Evaluate a double integral over a bounded region

I have the double integral $$\iint (x^4+y^2)dxdy$$ for the bounded region $y=x^2$ and $y=x^3$ Is this simple as integrating with respect to x, followed by y with $x^3$ and $x^2$ as limits? So i ...