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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290 views

Graph the integrand and use geometry to evaluate the integral

Graph the integrand and use geometry to evaluate the integral. $$\int_{-3}^{3}|x|+8\,dx$$
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2answers
148 views

Reduction formula tricky problem (Further Maths: F3)

$\int \:e^{ax}\cos ^n\left(x\right)dx$ I just cannot get it to reduce, I keep ending up with too many species in the next integral to use parts again. I have important Further Pure F3 exam in a month ...
2
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1answer
42 views

f whose second-order partial derivatives are continuous

Let $f: \mathbb{R} ^2 \rightarrow \mathbb{R}$ whose second-order partial derivatives are continuous, if $\int_C ( \frac{∂f}{∂y}dx - \frac{∂f}{∂x} dy ) = 0 $ for any closed curve simple $C$ then $ ...
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1answer
33 views

Maclaurin series and taylor

Im trying to find the first four terms of Maclaurin series of $\space0.15t^2$ and evaluate $$\int_0^1 e^{-0.15}t^2 dt $$ Please this is a revision question. How do i go about it?
3
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2answers
90 views

How to solve the integral $\int_0^1\frac{\cos x \ln x}{\sqrt{x}}$?

I am trying to solve the following integral: $\int_0^1\frac{\cos x \ln x}{\sqrt{x}}$ On wolfram-alpha I get the approximated value: -3.92203 Can anyone help me? Thanks in advance!
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4answers
241 views

Can someone help me to evaluate this integral:

Can someone help me with evaluating this integral: $$\int_{1}^{2} \frac{2x^2-1} {\sqrt{x^2-1}}\, dx$$ I tried using integration by parts, integration by substitution....but nothing...
3
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2answers
43 views

An integral I got on my midterm exam

Derive a recursive formula for the integral $I(n) = \int_0^1x^{m}\ln^{n}(x)\,dx$ and then solve the integral for $m = 0$. I have tried using partial integration as follows: $$ u = \ln^{n}(x) ...
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0answers
55 views

Prove the following: $\int_{a}^{b}|f\left(t\right)|dt\leq\left(b-a\right)\int_{a}^{b}|f'\left(t\right)|dt$

I have this homework question and I'm in need of some assistance: "Let there be a function $f:\left[a,b\right] \rightarrow\mathbb{R}$ continuously derivatable (every derivative is continuous), and ...
2
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0answers
118 views

About differentiation under the integral sign

I would like to ask something related to the application of the differentiation under the integral sign (Leibniz rule) given by ...
2
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0answers
32 views

Numerical Integration of Highly Oscillatory Integral with Misbehaving Derivatives

I'm attempting to numerically handle an equation of the following form: \begin{equation*}f: x \rightarrow \int_{0.00001}^{2} d\omega e^{i \omega x} f(\omega)\end{equation*} where $f(\omega) = ...
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2answers
32 views

Indefinite Multiple Integration

In multivariable calculus, definite integrals with multiple variables seem routine. However, I have not seen any example of an indefinite multiple integral. In fact, it seems as if limits of ...
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1answer
61 views

Find the area of a quadrangle with green´s theorem

How can i prove using Green´s Theorem that the area of a quadrangle with coordinates $(x_i,y_i)$ , $i=1,...,4$ is: $$A = \sum_{i=1}^{4} (x_{i}y_{i+1}-x_{i+1}y_{i})$$ I prove this with some examples ...
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0answers
31 views

Integrating over a probability distribution function

Say you have a probability distrubution fuction $P(x)$. Because this is a probability distribution, the following should be true; $$\int_{-\infty}^\infty P(x) \space dx = 1$$ Now, if you were to have ...
1
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1answer
68 views

Calculate $ \frac{\partial f}{\partial x} (x,y) $ [closed]

Calculate $ \frac{\partial f}{\partial x} (x,y) $ of $$ f(x,y) = \int_{x^2}^{y^2} e^{-t^2}\, dt$$
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1answer
35 views

How can I attack $\int_{-\infty}^\infty \frac{x}{[(x-vt)^2+a^2]^2} dx$?

I want to evaluate $$\int_{-\infty}^\infty \frac{x}{[(x-vt)^2+a^2]^2} dx$$ I know that $$\int \frac{1}{(\xi^2+a^2)^2} = \frac{\xi}{2a^2(\xi^2+a^2)}+\frac{1}{2a^3}\arctan\frac{\xi}{a}$$ Using ...
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vote
1answer
44 views

Finding general solution of first order DE's using integrating factor

I am asked to find the general solution of $$R\frac{dq(t)}{dt}+\frac{q(t)}{C}-V_0=0$$ I re-arrange so it is in the correct format. $$\frac{dq(t)}{dt}+\frac{1}{CR}\cdot{q(t)}=\frac{V_0}{R}$$ ...
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1answer
77 views

Evaluate the integral using principal value and complex analysis

I need to find the value of the integral: $\int_{-\infty}^{\infty} \frac{sin^2x}{x^2}dx $ Right now progress: Because the value of $\frac{sin^2x}{x^2}$ is convergent, the integral will be equal ...
3
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1answer
69 views

Why is $\int |e^{ix}|^2 dx = x + C$?

Quick question: Wolfram Alpha tells me that $$\int |e^{ix}|^2 dx = x + C$$ Why is that?
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0answers
26 views

How to find new region of integration after changing variable?

I'm not quite sure how to go about finding the new region of integration. The evaluation of an integral over a region D can be done by changing variables and integrating over a new region and ...
6
votes
2answers
171 views

Integral $\int_0^\infty \frac{y^{n}}{(1+y^2)^2}dy$

I wishing to calculating the integral$$ I:=\int_0^\infty \frac{y^{n}}{(1+y^2)^2}dy \qquad n=0,1,2 $$ I am looking for real analytic solutions thanks, the closed form is cosecant function so it is ...
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4answers
81 views

How to find the integral $\int4^{-x}dx$?

What approach would be ideal in finding the integral $\int4^{-x}dx$?
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0answers
28 views

Multivariable Calculus question about integrability

Let $U\subset \mathbb R^n$ included in rectangles $R,S$. Let there be function $f: U \rightarrow \mathbb R$ and let $f_R: R \rightarrow \mathbb R$, $f_S: S \rightarrow \mathbb R$ be $f$'s expensions ...
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1answer
29 views

Volume of a frustum with arbitary base area.

How do I find the volume of the frustum of a cone which has base area $A_o$, top area $A_t$ and height $h$? I am able to do this for circular and square bases, but unable to figure it out for ...
2
votes
1answer
87 views

Triple integral in spherical coordinates

I'm trying to evaluate the triple integral $\int\int\int_B\frac{dV}{\sqrt{x^2+y^2+z^2+3}}$, where $B$ is the ball of radius $2$ centered at the origin. Both the integrand and the nature of $B$ ...
3
votes
1answer
240 views

Integrate $\int_{-\infty}^{\infty}\exp\left(-\frac{\pi^2t(2x+1)^2}{2c^2}\right)\cos\left(\frac{(2x+1)\pi y}{c}\right)\exp(-2\pi i kx)dx$

By the poisson summation formula we have: $$\frac{1}{c}\sum\limits_{k=-\infty}^{\infty} \exp\left(-\frac{\pi^2t(2k+1)^2}{2c^2}\right)\cos\left(\frac{(2k+1)\pi ...
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vote
2answers
40 views

How to prove that this function is continuous?

If $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ is continuous on the rectangle $R=[a,b] \times [c,d]$, prove that the function $g(x) := \int\limits_{c}^{d} f(x,y) dy$ is continuous on $[a,b]$. Thanks in ...
2
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1answer
67 views

Integration by parts with log function [closed]

$$\int2x^3\ln(1-x^2)\;dx$$ How do I integrate this equation? Thanks.
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0answers
88 views

limit of p norm as p goes to 0!

Suppose we have a measure $\mu$ and a space $X$ such that $\mu(X)=1$, and a function $f \in L^r$ for some $r > 0$, where $L^r$ is defined in the usual way even for numbers less than $1$. Show ...
3
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0answers
35 views

How do I tackle this integral: $\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$? Is my solution correct?

I want to solve the following integral: $$\int_{-\infty}^\infty k\cdot |Ae^{-a|k-k_0|}|^2dk$$ I did the following: Substitute $\gamma(k) = k-k_0 \Leftrightarrow k = \gamma + k_0;~\gamma(\pm\infty) = ...
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1answer
69 views

Using complex variables to evaluate real trig integral

$$\int_0^{2\pi}\frac{\cos(2\theta)}{1-2a\cos(\theta)+a^2}d\theta, -1< a < 1$$ I did the substitution $z = e^{i\theta}$ and then got $\cos(2\theta) = 1/2(z^2+1/z^2)$, so ...
1
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1answer
38 views

Examples of functions with values in distributions

What is an example of a function in $L^p((0,T);\mathcal{D}'(\mathcal{R}))$? I ask this because the Majda-Bertozzi book on Incompressible flow deals with vortex sheet initial data $\omega(t,\cdot)\in ...
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1answer
82 views

Integrating around pie-slice domain

We are asked to show $$\int_0^{\infty}\frac{\log(x)}{x^3+1}dx=-\frac{2\pi^2}{27}$$, and $$\int_0^{\infty}\frac{1}{x^3+1}dx=\frac{2\pi}{3\sqrt{3}}$$ By integrating around a pie slice with angle ...
1
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1answer
85 views

Calculate integral using the residue theorem

I want to calculate the integral $$\int_C {{z^2-2z}\over{(z+1)^2(z^2+4)}}dz$$, where $C=\{z:|z|=4\}$ I want to use the Residue theorem to tackle this integral. Now, $f(z)$ has a pole of degree 2 at ...
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0answers
31 views

Elementwise normal to vector of unknowns and non-defined matrix multiplications

I wonder about statement (1). It is given that u is a column vector, A and B are constant, symmetric, square matrices of such size that the expressions on the left hand side of (1) are well defined. ...
2
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1answer
32 views

Least Squares Method Confusion

I'm learning about the Least Squares method. An exercise I am doing is "Find the constant c that makes the expression $$\int_{0}^{1} (e^x - cx)^2 dx$$ a minimum " Though, i'm not sure how to ...
1
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1answer
77 views

Show that $\lim_{x\to\infty} \frac{1}{x} \int_0^x f(x) \ dx = \gamma$.

The Assignment Let $f: [0,\infty) \rightarrow \mathbb{R}$ be a function which is integrable on the intervall $[0,x] \ \forall x > 0$ and $\lim_{x\to\infty} f(x) = \gamma \in \mathbb{R}$. ...
2
votes
1answer
62 views

Systematic method to change the order of integration in multiple integrals

In many examples of computation of multiple integrals, it is necessary to change the order of integration to achieve the computation. For example, $I=\int_0^1\int_y^1 \cos(x^2)\ dx\ dy$ can be ...
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1answer
54 views

Polynomial Basis with zero integral

I have a question and please I need help. First all, this is the context: Let $p : T\rightarrow\mathbb{R}$ a polynomial of degree $\leq k$ with null measure condition onto T, that is $$\int_T p\ =\ ...
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2answers
48 views

Integration with trigonomitry

Hi I was trying to find the area between the following curves (below) however I am unsure how to continue from the trigonometry which gets presented: Curves: $$y = 2\sin(x)\\y = ...
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2answers
36 views

Integrating two exponentials produces a cosine integral? Can somebody explain?

I discovered the following conversation that I do not understand. It reads: $$\int_{-U_1}^0 {(\frac {u_1} {U_1}+1)e^{-j\omega_1u_1}}~du_1+\int_0^{U_1} {(-\frac {u_1} {U_1}+1)e^{-j\omega_1u_1}}~du_1 = ...
3
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2answers
112 views

How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$ [duplicate]

Let $f$ be $C^1$ in $[-\pi, \pi]$ and satisfies $\int_{-\pi}^\pi f(x)dx=0$, periodic boundary condition. Then, prove that $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$. I try to prove ...
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vote
2answers
56 views

Find general solution of first order DE using integrating factor

I have the equation $$R\frac{dq(t)}{dt}+\frac{q(t)}{C}-V_0=0$$ And am asked to find the general solution using the integrating factor. I am a bit confused as I have been shown two ways to do it. ...
3
votes
4answers
153 views

How to find the integral for $\int 2^{\sin{x}}\cos{x}\;\mathrm{d}x$?

What would be the ideal approach in finding the integral for: $$ \int 2^{\sin{x}}\cos{x}\;\mathrm{d}x $$
3
votes
1answer
54 views

When $\int_{0}^{\infty}f(x)dx=\sum_{n=0}^{\infty}\int_{n}^{n+1}f(x)dx$?

Is the following always true? (i.e. if both converges, limits are equal; if one diverges, the other must diverge; EXCLUDE the case where the limit keeps "jumping") $$ ...
6
votes
2answers
190 views

Evaluation of Integral $ \int\ln(\tan x)dx$

Evaluation of Integral $\displaystyle \int\ln(\tan x)dx$ $\bf{My\; Try::}$ Given $\displaystyle \int\ln(\tan x)dx = \int \ln(\sin x)dx - \ln (\cos x)dx$ Now Using $\displaystyle \sin x = ...
0
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0answers
32 views

Contour integral going to zero on a limit

I've been asked to prove the following; $$\lim_{R \rightarrow \infty}\int_{C_R} \frac{z^2 + 8z + 42}{(z^2+4)(z^2-4z+5)}dz=0$$ Given that $C_R$ is a circle of radius $R$ centered at $0$. I thought ...
3
votes
1answer
109 views

Fundamental Theorem of Calculus when Integrand is a Function of the Bounds

I have a function, $$F(r) = \int_0^r |c x^2 + {(2 a + b - 4 a r - 3 b r - 2 c r) x^2\over2 r} + b x^3 + a x^4| dx$$ a, b and c are constants. I want to determine r such that $f=F'(r) = k$. ...
3
votes
5answers
94 views

How to integrate $\frac{\sqrt{x}}{1-\sqrt{x}}$?

How to integrate $\frac{\sqrt{x}}{1-\sqrt{x}}$? I tried by using integration by parts, but always got sucked. Should be very easy...
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0answers
22 views

Volume properties

How to go about proving this? Let $\Omega_1 \subset \Omega_2 \subset \mathbb{R}^n$ be two bounded regions such that the boundaries of both regions are sets of measure zero. Show that the volume of ...
1
vote
1answer
47 views

Integrable Manifolds

I'm trying to understand why the line of slope y passing through (x,y) is an integral manifold. My intuition tells me that there exists a point in the slope field where the distribution cannot be ...