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
votes
3answers
26 views

Show that $ f(x) = A.exp(2x) $ if $ f'(x) = 2f(x) $

Show that $ f(x) = A.exp(2x) $ if $ f'(x) = 2f(x) $ for some $ A \in \mathbb{R} $ Is it sufficient to say that if the derivative of a function contains itself, then it must be the exponential ...
0
votes
0answers
38 views

MAple 17 won't evaluate my integral

I type this into maple and it won't evaluate it: $$ \int_{0}^{1}\pi ((-y^4+1)^2-(1-y)^2) dy $$ I've also tried $$ evalf(\int_{0}^{1}\pi ((-y^4+1)^2-(1-y)^2) dy)$$ It just returns for both cases $$ ...
0
votes
1answer
22 views

Balancing the area of curves - integration

A linear equation needs to be found $y=ax+b$ with a slope of a maximum value of $10$ degrees that will balance out the area above the line and below the line for the function ...
0
votes
1answer
24 views

An application of Greens's theorem

Apply Green's theorem to prove that, if $V$ and $V'$ be solutions of Laplace's equation such that $V=V'$ at all points of the closed surface $S$, then $V=V'$ throughout the interior of $S$. ...
1
vote
2answers
53 views

How to compute $\int_0^1 \frac{x-1}{\ln(x)} dx = \ln(2)$? and $\int_0^\infty \ln(t) e^{-t} dt $?

$\int_0^1 \frac{x-1}{\ln(x)} dx = \ln(2)$ First i try $\ln(x)=t$ so that $\frac{1}{x} dx =dt$ then integral becomes \begin{align} &\int_{-\infty}^{0}\frac{e^t-1}{t} (e^t dt) = - ...
7
votes
3answers
175 views

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

$$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$ I have difficulty to evaluating above integrals. First i try the subsititue $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...
-3
votes
2answers
38 views

Integral of $f(x)= x\sin(n \pi \cdot \frac{x}{L})$ [closed]

How do I go about integrating $x\sin\left(n \pi \cdot \frac{x}{L}\right)$ with respect to $x$? (which methods etc?) Thanks in advance for tips/advice/solutions.
0
votes
2answers
43 views

Definite integral from -1 to 0 [closed]

How would I evaluate this definite integral $$ \int_{-1}^{0}\tan x dx- \int_{-1}^{0}\sin^2 x dx $$ All i need to know is what to do when an integral is on an interval of -1 to 0. I could do this ...
3
votes
0answers
50 views

Strange triple integral of an inverse function

Let $$ \Omega(a, b, c) = \min\left\{\theta\ge0\ \text{s.t.}\ \tan(a\theta) + \tan(b\theta) + \tan(c\theta) = 1\right\} $$ What is the value of the following integral $$ I = ...
0
votes
0answers
26 views

Using the general slicing method to find the volume of a semi-circle whose cross sections are squares.

In finding the volume of a solid, described below, I was close in finding the equation, but neglected a coefficient. Please see the question below. Use the general slicing method to find the volume ...
1
vote
1answer
36 views

How to integrate hydrostatic force on a two dimensional shape?

I'm so confused this question is very different from the other hydrostatic force questions and I think I am misunderstanding the question. I am primarily concerned with 15 because I somehow managed ...
3
votes
5answers
578 views

Don't understand the Fundamental Theorem of Calculus

If $f$ is continuous on $[a, b]$ and defining $$ F(x) = \int_a^x \! f(t) \, dt $$ for $x \in [a, b]$, then $F'(x) = f(x)$ for $x \in (a,b)$. I don't understand what function the variable ...
0
votes
0answers
32 views

how to use complex integration to calculate $\int_0^{\pi}(1/a+\cos(x))dx$?

I have so far replaced $dx$ by $1/zi \ dz$, but I don't know how to deal with $\cos(x)$
1
vote
1answer
35 views

Definite integral including natural log, cosine, and hyperbolic sine

Here is an integral question I have, I am solving some other problems like this but I am stumped on this one: $$\int_0^{\pi+1}\frac {\ln(\cos(x+1))}{\sinh(x^2)}dx$$ I used some methods such as ...
-1
votes
0answers
24 views
0
votes
1answer
44 views

Find the volume of the solid in $\Bbb R^3$

I need to find the volume of the solid in $\Bbb R^3$. It is bounded by the following: $y=x^2$, $x=y^2$, $z=x+y+21$ and $z=0$. I known that the volume is expressed as follows: $$\iiint 1 \, dV$$ I ...
1
vote
4answers
46 views

Area between three lines/curves

I know this is a very elementary question but I can't make out the answer from the other posts I found in my search. These are three lines, I need to find the area enclosed by them. how do I go ...
3
votes
1answer
54 views

Solving an integral with trig substitution

I'm looking to solve the following integral using substitution: $$\int \frac{dx}{2-\cos x}$$ Let $z=\tan\frac{x}{2}$ Then $dz=\frac 1 2 \sec^2 \frac x 2\,dx$ $$\sin x=\frac{2z}{z^2+1}$$ $$\cos x ...
0
votes
1answer
9 views

Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ (Riemann integral)

Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ where $L(F)$ is the supremum of the lower sums of the Riemann integral. I tried to by contradicction ...
0
votes
1answer
20 views

Applying the fundamental theorem of Integration

We know from the fundamental theorem of Integration that for a continuous function $ f:[a,b] \rightarrow \mathbb{R} $ with antiderivative $ f:[a,b] \rightarrow \mathbb{R} $ we have that $ ...
3
votes
2answers
63 views

how can I show this integral diverges?

I want to show $E(T_a)=\infty$ $$E(T_a)=\int_0^{\infty}{{x|a|}\over\sqrt{2\pi}}x^{-3/2}e^{-a^2/x}dx$$ to show this I need to show this integral diverges. I know gamma function that $$\Gamma ...
1
vote
2answers
29 views

Iterated integrals in general ( and double integral )

$f:[0,1]\times [0,1]\to\mathbb R,$ defined by $$f(x,y)= \begin{cases}1,\quad \ \ y\in\mathbb R\text{\\}\mathbb Q\\2x,\quad\text{otherwise}\end{cases}$$. $1.1$: $\int_0^1f(x,y)dx$ exists for every ...
4
votes
2answers
46 views

Writing integral in terms of distributions

EDIT (now asking how to write $F$ as distributions, instead of writing the integral in terms of distributions): Let $F$ be the distribution defined by its action on a test function $\phi$ as ...
6
votes
6answers
98 views

Two apparently different antiderivatives of $\frac{1}{2 x}$

What is right way to calculate this integral and why? $$ \int\frac{1}{2x}\text dx $$ I thought, that this substitution is right: $$ t = 2x $$ $$ \text dt = 2\text dx $$ $$ \frac{\text dt}{2} = ...
1
vote
1answer
27 views

Integral equation involving Planck radiation formula

I am stuck in solving the following integral equation: $$\sigma T^4=\pi\int_{\lambda_0}^{\lambda_1}d\lambda W_{\lambda,T}$$ where: ...
0
votes
1answer
27 views

what is the Convolution the expression [closed]

what is the convolution of the expression, $\ x(t)*y(-t)\ $ I want to apply it to be written in integral form.
0
votes
1answer
47 views

Fourier coefficients of a symmetric function in $\pi$

I want to show that the Fourier coefficients $\int_{-\pi}^\pi e^{ij \lambda} f'(\lambda) d \lambda$ of the derivative of a continuously differentiable function $f: [ - \pi, \pi] \rightarrow ...
0
votes
0answers
30 views

convergence and holomorphic function - explanations and proofs

I haven't had analysis for a long time and I've forgotten plenty. Could you please explain me and prove that the following converges: $$ \sum_{n \geq 1}e^{-n^2t\pi}, t>0 c $$ and explain why the ...
-1
votes
0answers
45 views

Could you help me an integral question? [closed]

Prove that $$ e \int_0^{1/e} (1-x)^{-1/x}\,dx>3.039 $$ where $e=2.71828\dots$ Thanks a lot.
0
votes
2answers
63 views

How to integrate $x/\sqrt{1-x^2}$?

Could somebody please tell me where I made a mistake? I want to integrate \begin{equation*} \int_a^b\frac{x}{\sqrt{1-x^2}}dx. \end{equation*} As far as I know the subsitution $u=1-x^2$ works, but ...
0
votes
1answer
22 views

vector field using green's theorem+other integration

So am I supposed to be using green's theorem for the first question, but where I'm confused is that there are three variables if I do, dx dy dz (I haven't learn how to use green's theorem for 3 ...
1
vote
3answers
99 views

Trigonometric Substitution in $\int _0^{\pi/2}{\frac{ x\cos x}{ 1+\sin^2 x} dx }$

Evaluate $$ \int _{ 0 }^{ \pi /2 }{ \frac { x\cos { (x) } }{ 1+\sin ^{ 2 }{ x } } \ \mathrm{d}x } $$ $$$$ The solution was suggested like this:$$$$ SOLUTION: First of all its, quite obvious to have ...
4
votes
0answers
37 views

$\int_{-\pi/2}^{\pi/2} \cos(a \cos\theta) e^{im\theta} e^{-ib\sin\theta} \mathrm{d}\theta $ Integration

I am struggling to find the integration of the expression below, $$\int_{-\pi/2}^{\pi/2} \cos(a \cos\theta) e^{im\theta} e^{-ib\sin\theta} \mathrm{d}\theta $$ where $a$ and $b$ are arbitrary constant ...
0
votes
4answers
60 views

Compute Integral of $\int_0^1 e^{\sqrt{x}}dx$ using integration by substitution

Compute Integral of $$\int_0^1 e^{\sqrt{x}}dx$$ using integration by substitution. I will start from: $\int_0^1 e^{\sqrt{x}}dx$ let $u= \sqrt{x}$, $du = -dx\rightarrow dx=-du$ Now I will start ...
1
vote
0answers
29 views

Integration of gaussian divided by square root of -log(1-x) - does the Meijer G function help me?

After some modelling of my data I came to the following integral: $$ \int_0^{1}\dfrac{exp{\left(-\dfrac{\left(x-\mu\right)^2}{2\,\sigma^2}\right)}}{\sqrt{-\log{(1-x)}}} $$ I cannot solve it, and ...
0
votes
2answers
25 views

evaluating the double integral

I tried to calculate $\int _0^9 dx\:\int _{-\sqrt{x}}^{\sqrt{x}}\:y^2dy$ which yielded $c$ as in this integral has no particular value...when I plot the graphs for it's D however, a certain area does ...
0
votes
2answers
58 views

Understanding limits that occur in integrals during Feynmann integration

When we use Feynman Integration, how do we decide the final constant of integration? For example, in this problem: $$ f(a) = \int_{0}^{1} \frac{\arctan(ax)}{\sqrt{1-x^2}} dx$$ After ...
0
votes
0answers
44 views

Feynman Integration

Could somebody please recommend some places to learn Differentiating under the Integral sign? I need to learn this technique as a lot of Integrals can be solved using this. Many thanks!
0
votes
1answer
36 views

How to calculate the shielding time and determine the time step

The problem is illustrated as follows. A shielding plate scans over a target plate at a constant speed $v_{scan}$ and dynamically shadows the target plate to adjust the exposure time of the light ...
4
votes
0answers
46 views

Difficult definite integral involving Bessel function in the denominator

I came across an integral involving Bessel function in the denominator, derivatives of Bessel functions and complex argument.The equation yields: $$ \int_0^1 ...
0
votes
0answers
32 views

Calculation of a volume integral

Calculate the volume integral $$\int_V z^2r^{-3}e^{-r^2} \, {\rm d}V$$ where $r=\sqrt{x^2+y^2+z^2}$ and $V$ is the whole of $\mathbb{R}^3$. So I want to integrate $$\int_V z^2r^{-3}e^{-r^2} \, {\rm ...
1
vote
1answer
63 views

$\int_{0}^{\frac{\pi}{4}} e^{\sec x} \frac{\sin( x + \frac{\pi}{4})}{(1 - \sin x) \cos x}\, dx$?

How do I find the value of $$ \int_{0}^{\frac{\pi}{4}} e^{\sec x} \dfrac{\sin\Big( x + \dfrac{\pi}{4}\Big)}{(1 - \sin x) \cos x} \;\mathrm{d}x $$
-3
votes
0answers
20 views

Calc 2 Work Problem [closed]

In a steam engine the pressure P and the volume V of steam satisfy the equation PV^1.4=k, where k is a constant. Use the information given to calculate work done (in ft-lb) by the engine during a ...
2
votes
1answer
55 views

Help with an Inverse Trigonometry Integral 2

Evaluate $$\int^{1/{\sqrt{3}}}_{-1/{\sqrt{3}}} \frac{x^4}{1-x^4}\cos^{-1}\frac{2x}{1+x^2} \mathrm{d}x\\= \frac{\pi}{a}\ln(b+\sqrt{c}) +\frac{\pi^{d}}{e} - \frac{\pi}{\sqrt{f}}$$ Then Find ...
-1
votes
0answers
38 views

Solve $\int_{0}^{\frac{\pi}{2}}\ln^{2}(\sin(x)) \ \mathrm dx$ [duplicate]

I recently had this integration problem on a test and could not solve it any way I tried. How do I do this? $$ \large\displaystyle\int_{0}^{\frac{\pi}{2}}\ln^{2}(\sin(x))\mathrm dx$$
-2
votes
0answers
51 views

Calculus 2 Work Problem [closed]

In a steam engine the pressure P and the volume V of steam satisfy the equation $PV^{1.4}=k$, where $k$ is a constant. Use the information given to calculate work done in ft-lb by the engine during a ...
2
votes
1answer
78 views

Integral with Logarithms

$$\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\log(\sin(x)) \ dx } = \dfrac { \pi { \ln}^{ A }(B) }{ C } -\dfrac { { \pi }^{ D } }{ E } $$ $$$$ This was one solution, but it went completely ...
4
votes
1answer
41 views

Simple Derivation of Functional Equation Question (L'Hospital's Rule)

First, the question is: $f$ is a differentiable function and $f : R \rightarrow R$ $xf(x)-yf(y)=(x-y)f(x+y)$ $f'(2x)=?$ My approach for problem is using L'Hospital's rule: $$ ...
1
vote
1answer
39 views

$\int \dfrac{1}{4x^2+1} \ dx $ - why is it $\dfrac{1}{2} \arctan(2x)+c$ not $\arctan(2x)+c$?

$\int \dfrac{1}{4x^2+1} \ dx $ - why is it $\dfrac{1}{2} \arctan(2x)+c$ not $\arctan(2x)+c$? I've been looking at my formula booklet which gives the integral as: $$\int ...
3
votes
2answers
54 views

Evaluating the complex integral $\int_{-\infty}^\infty \frac{\cos(x)}{x+i}\,dx$

I stumbled upon this particular integral a few minutes ago, and I have no idea how to go about it : $$\int_{-\infty}^\infty \frac{\cos(x)}{x+i}\,dx$$ I looked up on the internet and I managed to ...