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
62 views

System of differential equations: where did I got wrong?

I have the system: $$x_1'=2x_1-x_2+e^{2t}\\ x_2'=4x_1+2x_2+4$$ So I searched for the homogeneous solutions and got: $$X_H=c_1e^{2t} \left(\begin{matrix} \cos 2t \\ 2\sin ...
1
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3answers
26 views

Integral with functions as limit

How can I solve integrals like this : $$ \int_{g(x)}^{f(x)} u(t) dt $$ I think it has a general formula to solve it. Thanks for your help.
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2answers
39 views

Why does $\int_{0}^{\infty} \lambda e^{(t-\lambda)x} dx$ exist?

According to my textbook (Grinstead and Snell's Introduction to Probability), $\int_{0}^{\infty} \lambda e^{(t-\lambda)x} dx = \frac{\lambda}{\lambda-t}$. But it seems to me like there are two ...
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1answer
32 views

Transformation of an improper integral of the square root of a rational function

Why is this true? $$\int_{-\infty}^{\infty}\sqrt{\frac{1}{(t^2-1)^2}-\frac{(n+1)t^{2n}}{(t^{2n+2}-1)^2}}dt=4\int_{0}^{1}\sqrt{\frac{1}{(t^2-1)^2}-\frac{(n+1)t^{2n}}{(t^{2n+2}-1)^2}}dt$$ I know that: ...
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0answers
18 views

Volume of a solid between 2 hyperbolic planes

I want to calculate the volume of the solid between the two planes : $$z=3x^2+3y^2 \qquad \text{ and } \qquad z=4-x^2-y^2.$$ Do my answer is right!? \begin{align} \int_{0}^{2\pi} \int_{0}^{1} ...
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1answer
30 views

A piecewise function inserted into an integral

Let $ f : \mathbb{R}^{+} \times \mathbb{R}^{+} \to \mathbb{R}$ be a nice function, and let's say we define a function $g: \mathbb{R} \times \mathbb{R}^{+} \to \mathbb{R}$ as: $g(x,t) := f(x,t)$ for ...
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1answer
22 views

Riemann left,right and midpoint sums

Find left, right and midpoint Riemann sums for $\displaystyle\int_{1}^{2} \frac{1}{x} dx$ $P = 2, \frac{5}{2},3,4 $ Using: $f(x_i)\Delta x$ Please check my work: ...
1
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1answer
28 views

Computing the derivative: $\frac{\partial}{\partial x} \left\{ \int_0^t \int_{x - t + \eta}^{x + t - \eta} F(\xi,\eta) \,d\xi\, d\eta \right\}$

Let's say that $F$ is a nice well-behaved function. How would I compute the following derivative? $$\frac{\partial}{\partial x} \left\{ \int\limits_0^t \int\limits_{x - t + \eta}^{x + t - \eta} ...
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1answer
63 views

Finding Velocity and Distance Formula from Integral of Acceleration

Working on a homework problem here for integration that has two parts: (1) Suppose that an object of mass m is initially held at rest, then is released and acted upon by a constant gravitational ...
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1answer
18 views

How do i calculate the number of subintervals n in Midpoint method?

I want to calculate the least error (o) in order to obtain the exact answer for integration using the midpoint method. However I am having trouble doing so since i was given a functions whose second ...
2
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2answers
78 views

Integrate $\int \frac {\ln x}{x^2}$ by U-Substitution

I've got a question regarding "improper" u-substitution integrals. How would we solve $\int \frac {\ln(x)}{x^2}dx$ using the u-substitution $u = \ln(x)$? I know that it might be easier to solve it ...
0
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1answer
27 views

Evaluate $ \int_0^a\int_x^a\frac{y^2\,dy\,dx}{\sqrt{x^2+y^2}}$ by changing order of integration

The question is: By changing the order of integration in the integral $I = \int_0^a\int_x^a\frac{y^2\,dy\,dx}{\sqrt{x^2+y^2}}$ show that $I=\frac{1}{3}a^3\,\ln(1+\sqrt{2})$ My attempt so far: So, ...
2
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4answers
86 views

calculate $\int_0^\infty x^n\sin(2\pi \ln(x))e^{-\ln^2(x)} \mathrm{d}x$

I have to show, that $$\int_0^\infty x^n\sin(2\pi \ln(x))e^{-\ln^2(x)} \mathrm{d}x=0$$ and I know, that $\int_{-\infty}^{\infty} \sin(2\pi t)e^{-t^2} \mathrm{d}t=0$. So, I thought of substitution: ...
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2answers
49 views

Odd function in an integral

I have the following integral \begin{equation} \int^{2\pi}_{0}(16 \sin \phi \cos^2\phi-4\sin^{2}\phi+8\sin\phi)d\phi = -4\pi \end{equation} It says that is odd function of $\phi$. Can anyone ...
2
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2answers
80 views

Convergence of $\int_1^\infty \frac{dt}{t^4+t^2+1}$

I have to calculate $\displaystyle\int_1^\infty \frac{dt}{t^4+t^2+1}$ and i have as a hint: Divide by $t^2+t+1$ Then I get that $(t^2+t+1)(t^2-t+1)= t^4+t^2+1$ I could do this by partial fractions but ...
0
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2answers
28 views

Evaluate the improper integral $\int_0^{\pi/2}(\sec^2x-\sec x \tan x)dx$

Evaluate the improper integral $\int_0^{\pi/2}(\sec^2x-\sec x \tan x)dx$ I got $\int_0^{\pi/2}(\sec^2x-\sec x \tan x)dx=\int_0^{\pi/2} \frac{1-\sin x}{\cos^2x}dx$ The singularity occurs at the ...
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2answers
25 views

Calculating $\iint_D(x^3+y^3)dxdy$ for $D=\{(x,y)~|~0\le x,y\le1\}$

How do I calculate $\iint_D(x^3+y^3)dxdy$ for $D=\{(x,y)~|~0\le x,y\le1\}$? Specifically what I don't get is what do I let the upper bound of $x$ and lower bound of $y$ be?
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1answer
50 views

What's wrong in my way to calculate $\int_0^a\frac{1}{\sqrt{a^2-x^2}}\,dx$?

I want to calculate this integral $$ \int_0^a\frac{1}{\sqrt{a^2-x^2}}\,dx. $$ I did like: Let $\frac xa = \sec(u)$ then $dx=a \sec(u) \tan(u)du$ When I do this, I recognize something wrong: That ...
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2answers
50 views

Solve exponenital integral equation

$$\frac{1}{\sqrt{2\pi}\sigma_1 }\int_x^\infty\exp(-\frac {t_1^2-1}{2\sigma_1^2})dt_1 + \frac{1}{\sqrt{2\pi}\sigma_2 }\int_x^\infty\exp(-\frac {t_2^2-1}{2\sigma_2^2})dt_2 = a $$ $$\sigma_1 , \sigma_2 ...
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2answers
238 views

Finding mass of a sphere given density = $1-\rho^2$ and radius = 1

I'm currently working on what should be a relatively simple problem but I'm not sure if I'm right in my answer. This is for Calc3 homework that I'm just trying to get a grasp on the concepts before ...
-1
votes
1answer
61 views

Volume of the solid bounded by the cylinder and planes

Volume of the solid bounded by the cylinder $z = 1-x^2$ and planes, $x=y, y=0,z=0$ $$\int_{y=0}^1\int_{x=y}^1 (1-x^2) \, dx \, dy$$ I don't know if my upper and lower limits are correct. Tell me if ...
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3answers
49 views

Evaluate iterated integral by changing to polar coordinates

$$\int_0^{1/2}\int_0^{\sqrt{1-x^2}}xy\sqrt{x^2+y^2}\,dy\,dx$$ $x^2+y^2=r^2$ $$\int\int_0 r^3\cos\theta \sin\theta|r|\,dr\,d\theta$$ I don't know what $r =$ at line $x = 1/2$. I don't know value of ...
1
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1answer
51 views

What kinds of multiple integrals are “allowed”? e.g. $\int_0^a \int_0^a f(a)\,da\,da$

This is something I was thinking about when I was making an effort to catalog all the different types of multiple integrals. According to my current understanding, it seems like some are allowed while ...
2
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2answers
51 views

How would you differentiate this? I can't get anywhere

Let's say that $F$ is a nice well-behaved function. How would I compute the following derivative? $\frac{\partial}{\partial t} \left\{ \int_{0}^{t} \int_{x - t + \eta}^{x + t - \eta} F(\xi,\eta) d\xi ...
1
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2answers
25 views

Joint distribution functions and finding the bounds

I am trying to find the marginals of this function, however, I am not sure how to determine the limits of integration. The answer key says that for $f_1(y_1)$, the limit is from $y_1$ to 1, and for ...
10
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2answers
190 views

Closed form for ${\large\int}_0^\infty\frac{\arctan(x)\,\operatorname{arccot}(x+1)}{x}dx$

I'm looking for a closed form for this integral: $$I=\int_0^\infty\frac{\arctan(x)\,\operatorname{arccot}(x+1)}{x}dx.$$ Mathematica and Maple could not evaluate it symbolically. Numerically, ...
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1answer
38 views

Evaluate $\int_{-3}^{3} \int_{0}^{\sqrt{9-x^2}} \sin(x^2+y^2) \, dy \, dx$

I need to know if my result is correct : $$\int_{-3}^{3} \int_{0}^{\sqrt{9-x^2}} \sin(x^2+y^2) \, dy \, dx$$ So the region to integrate is $$R:=\left\{ (x,y) \mid 0<y<\sqrt{9-x^2} \text{ and ...
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4answers
51 views

Calculation of $\frac{1}{\sqrt{2\pi}\sigma(x)}\int_{-\infty}^{\infty}|u|\exp\left(-\frac{u^2}{2\sigma{^2}(x)}\right)\ du$

How to show that $$\dfrac{1}{\sqrt{2\pi}\sigma(x)}\int_{-\infty}^{\infty}|u|\operatorname{exp}\left(-\dfrac{u^2}{2\sigma{^2}(x)}\right)\mathop{du}=\sqrt{\dfrac{2}{\pi}}\sigma(x)$$ I think it has to ...
0
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2answers
76 views

Evaluate $\int_{0}^{1} \int_{3y}^{3} e^{x^2} \, dx \, dy$

I have to evaluate the integral $$\int_{0}^{1} \int_{3y}^{3} e^{x^2} \, dx \, dy.$$ Can you give me a hint because I can't figure it out how to integrate $\int e^{x^2} dx$.
2
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2answers
41 views

Volume between cone and sphere - First octant

Find the volume between $z=\sqrt{x^2+y^2}$ and the sphere $x^2+y^2+z^2=1$ that lies in the first octant using cylindrical coordinates. So I found the intersection and got $r=\frac{\sqrt2}{2}$. I know ...
11
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2answers
175 views

What's exactly the deal with differentials? (Confessions of a desperate calculus student)

So I don't know if I'm the only one to feel this, but ever since I was introduced to Calculus, I've had a slight (if not to say major) aversion to differentials. This sort of "phobia" started from ...
2
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1answer
34 views

Integration with trigonometric substitution using x=a.cos instead of x=a.sin

I was trying to integrate $ \int \frac{\sqrt{9-x^2}}{x^2} dx $. If I substitute $ x=3\sin{\theta} $ the result will be $ -\frac{\sqrt{9-x^2}}{x}-\sin^{-1}{(\frac{x}{3})} +C$, which is the correct ...
1
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0answers
25 views

Integral of product of $h_{x_{i}}(x)$ and $h_{x_{j}}(x)$, where $h_{c}(x) = \min(c,x)'$

I am trying to integrate the product of $h_{x_{i}}(x)$ and $h_{x_{j}}(x)$, where $ h_{c}(x) = \min(c,x)'$ as follows: $$ h_{c}(x) = [\min(c,x)]' $$ $$ \int_0^\infty ...
1
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1answer
27 views

Find $\int_0^{y^2}e^{-xy}dx$ using the Leibnitz integral

$$\int_0^{y^2}e^{-xy}dx$$ The Leibnitz integral rule states that: $$\frac{d}{dy}\int_{x_1}^{x_2} f(x,y)dx=\int_{x_1}^{x_2} (\frac{\partial f}{\partial y})dx$$ However I can't seem to get this to ...
0
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1answer
14 views

Fisher distance in regards to classification

I'm trying to understand what Fisher Distance actually is: Ruiz et al As well I am unsure how the writer of the paper has gotten from equation 4 to 5 by substituting in the activation function as ...
2
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3answers
60 views

Volume of sphere - order of integration

Can anyone point out my mistaken reasoning please? Question Use triple integral to find the volume of a sphere of radius $R$. Can do the following and it works fine: $$V = ...
1
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1answer
46 views

Improper integral $\int_{-\infty}^{+\infty}\frac{1}{a x^2 + bx + c} dx$, $a > 0,a x^2 + bx + c>0$

I have a question about improper integral. If you can help me , I appreciate that. If a > 0 and the graph of $y=a x^2 + bx + c$ lies entirely above the x-axis, show that $$ \int_{-\infty}^{+\infty} ...
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4answers
35 views

Integrate $\int{(2x+1)\sqrt{2x-1}}\,dx$ with manipulation

I need your help with $$\int{(2x+1)\sqrt{2x-1}}\,dx$$ The problem is the answer options are all manipulated. In the options, they combine $\frac{2}{189}(6-3x)^{7/2}$, $\frac{8}{45}(6-3x)^{5/2}$, and ...
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3answers
33 views

Integrate rational function

Can you help me to integrate: $$ \frac{(1-cos(x))}{sin^2(x)}$$ between $pi/2$ and $pi/4$ I've seen Wolfram's solution but was wondering if there was another way to do it. Thanks
3
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1answer
115 views

How does wolfram alpha compute this integral?

Wolfram alpha can find that $\int_1^\infty\frac{x^8-8x^7+24x^6-32x^5+19x^4-12x^3+17x^2-10x+2}{x^{12}-12x^{11}+56x^{10}-120x^9+82x^8+112x^7-182x^6-92x^5+381x^4-356x^3+170x^2-44x+5}dx=\frac{\pi}{2}$ ...
2
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1answer
70 views

What is happening in this integration?

I found in Peskin-Schroeder, while reading Quantum Field Theory. the following integration. $$\frac{1}{4\pi^2 r}\int_m^\infty \frac{se^{-sr}}{\sqrt{s^2-m^2}} = e^{-mr}$$ at the limit $r \rightarrow ...
1
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1answer
35 views

Area under curve fundamentals

I'm reading Stroud's Engineering Mathematics. A small excerpt from Stroud's book follows: Area under the curve_part1 Area under the curve_part2 I'm a software engineer and I know how to integrate ...
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0answers
32 views

Best integration method to integrate this oscilating function

What method of integration, in matlab, should i use ? integral ? quadgk ? gausslegendre ?? any ideas ? this function oscilates a lot. I used trapz, but is was not a good idea the link of images is ...
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3answers
115 views

Calculus problems involving motion

I've been working on the following problems, and I know how to integrate functions,but I do not know how to find the value of "c" in the examples below when finding the antiderivative. Any idea what ...
3
votes
2answers
105 views

Convolution of half-circle with inverse

I am trying to compute the function: $$f(\lambda)\equiv\int_{-1}^{1}\frac{\sqrt{1-x^2}}{\lambda-x}dx.$$ It arises as the convolution of the semi-circle density with the inverse function. When ...
1
vote
1answer
101 views

Differentiating a definite integral

I am unsure of how to differentiate the follow expression: $$ \frac{d}{dt} \int_\tau^t \Phi(t,\alpha)\cdot A(\alpha)\cdot x(\alpha) d\alpha $$ where $\Phi(t,\alpha)$ and $A(\alpha)$ are matrices.
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3answers
86 views

Integral with inverse trig functions and u substitution

I've been trying to find the integral of $\dfrac{dx}{x \sqrt{x^2-4}}$. Currently, following the example problem, I have it as $\dfrac{1}{2} S \dfrac{1\,du}{x \sqrt{4 [(\frac{x}{2})^2-1]}}$. From what ...
5
votes
1answer
124 views

Help integrating the transition probability of the Brownian Motion density function.

1. Problem: Given the Brownian Motion with Drift: $$ dx = \mu \, dt+\sigma \, dW $$ It can be shown that the transition density function is the following: $$ p(x, t) = \frac{e^{-\frac{(x-\mu ...
1
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1answer
63 views

Exchanging Series with Integrals: when is it possible?

In what cases is it possible to exchange between Series and Integral signs, namely performing the following operation? $$\sum\int F(x) \text{d}x ~~~~~ \to ~~~~~ \int\sum F(x) \text{d}x$$ What are ...
1
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1answer
89 views

Feynman problem on action

It is very weird for me that a newbie can ask a new (may be silly, sorry...) question but must have 50 reputation to comment. When I see a good question like this but have no answer what I have to ...