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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26 views

Integration U Substitution Involving e

Edit: The integral needs to be solved with the exact value of u given below. I am asked to solve the following indefinite integral: $\int\frac{e^{6x}}{\sqrt{9-e^{12x}}}dx$ By making the ...
0
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1answer
25 views

Integration by transformation

For $s\in [0,1]$, I can compute this integral, $$\int \frac{sx^2(1-x)}{1-sx^2}dx$$ by ignoring the denominator and computing $\int sx^2(1-x)dx$, as the difference is almost negligible. However, I am ...
1
vote
1answer
43 views

How to calculate the definite integral

How to do the below integral: $$\int_0^\infty \sqrt{x} e^{-x} dx = \frac{\sqrt{\pi}}{2} $$ I am guessing it somehow relates to $$ \int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi} $$
0
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1answer
49 views

What is the method to use the generalised Cauchy Integral Formula

Past Paper Question: a) State the generalized form of Cauchy’s integral theorem b)Evaluate $$\displaystyle f(z)=\int_{\gamma}\frac{z^2}{\biggr(z-\dfrac{\pi}{4}\biggl)^3} dz$$ where $\gamma$ ...
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0answers
47 views

Representation of a real function through a Fourier Transformation

I 'm trying to do some calculations regarding some differential equations and I came across an interesting way to express a real function through a double integral of the form: ...
2
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0answers
33 views

Would an integral defined using partitions of an interval into infinitely many intervals make sense?

In the definition of Riemann integral or Darboux-integral we first study partitions (or tagged partition) of the given interval determined by finitely many points. To each partition and a function $f$ ...
3
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3answers
96 views

The values of constants in the Equation.

$$ \frac{\int_0^{4\pi} e^t(\sin^6at+\cos^4at)\,dt}{\int_0^\pi e^t(\sin^6at+\cos^4at)\,dt}= L, $$ the question asks the value of $a$ and $L$. My friend solved it by differentiating, but i didn't ...
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2answers
53 views

number of distinct solution of Integral equation

Total number of distinct $0\leq x\leq 1$ for which $\displaystyle \int_{0}^{x}\frac{t^2}{1+t^4}dt = 2x-1$ $\bf{My\; Try::}$ Given $\displaystyle \underbrace{\int_{0}^{x}\frac{t^2}{1+t^4}dt}_{\geq ...
1
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0answers
33 views

Details on Proving that $\lim_{n \rightarrow \infty}\int_{-M}^M f(x) \cos (nx) dx=0$ Using Density of Step Functions

I was working on a question very similar to this post: Show that $\int_{-\pi}^\pi ~f(x) \cos (nx) \mathrm{d}\mu(x)$ converges to $0$ . I want to show that $\lim_{n \rightarrow \infty}\int_{-M}^M ...
2
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2answers
91 views

Prove that $\int_0^\infty \frac{\sin nx}{x}dx=\frac{\pi}{2}$

There was a question on multiple integrals which our professor gave us on our assignment. QUESTION: Changing order of integration, show that $$\int_0^\infty \int_0^\infty e^{-xy}\sin nx \,dx ...
2
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2answers
242 views

Changing the bounds of integration

I have a question that asks me to find the derivative of this integral, with out evaluation the intergral. $$\int_{\sin x}^{\cos x}\frac {1}{1-t^2}dt$$ I think I need to use U-substitution and the ...
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1answer
46 views

Having trouble evaluting error function integrals

I am trying to evaluate $$I = \int_1^{\infty } \left(\frac{\operatorname{erf}\left(a -b\log (x)\right)}{2 x^2}-\frac{\operatorname{erf}\left(a + b\log (x)\right)}{2 x}\right) \, dx$$ Let $\log (x) = ...
5
votes
3answers
120 views

Is the Riemann integral of a strictly smaller function strictly smaller?

We all know that if $f\leq{}g$ in $[a,b]$ then $$ \int_a^bf\,dx\leq\int_a^bg\,dx $$ now, imagine that we have $f<g$, is it true that $$ \int_a^bf\,dx<\int_a^bg\,dx $$
1
vote
1answer
51 views

On changing limits of integration when there are domain problems.

As an example say I have $$\int_{\pi/2}^\pi \frac{2}{1- \sin(2x)} dx$$ I would like to perform the substitution $2x = \arcsin(u)$ but I notice this would not be surjective on the interval given by ...
-1
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0answers
77 views

Very Complex Integral

I'm facing a hard time solving the below integral, and I can you a hand in here :) $$\int_a^\infty \text{erfc} \left( \sqrt{x} \right) \cdot \frac{1}{x} \cdot e^{-\left(\ln \left( \frac{x}{b} ...
0
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2answers
63 views

solve for y in $c = \int_{0}^{y} \sin^n(x) dx$ [closed]

I'm new to Mathematica so this could be an easy question : I'd like to solve for y in $$c = \int_{0}^{y} \sin^n(x) dx$$ where c and n are parameters; n could be large $n=100$; c could be small ...
2
votes
1answer
61 views

Does the integral converge?

$$\int_0^{\infty} \frac{\cos(x+1/x)}{\vert \ln (x) \vert ^p}\,dx, ~p \in \mathbb{R}$$ I tried to change $t=x+1/x$, but it get worse. I have no idea how to start.
0
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1answer
42 views

How does this integral go to the next step?

How did he manage to pull $1/4$ out and separate the bottom brackets in the following computation? $$\int\frac{dy}{(y+3)(y-1)}=\frac14\left(\int\frac{dy}{y-1}-\int\frac{dy}{y+3}\right)$$
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1answer
16 views

Volume bounded between to multivariable functions

Really would like some pointers on how to attack this! I understand how to find the integrand, but how do you get the bounds?. The correct answer is Choice B.
2
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1answer
26 views

Some of the Cauchy type integral property

Let $D$ - a simply connected bounded domain and $\phi(t) \in C(\partial D)$. Prove that $ \displaystyle \oint \limits_{\partial D} \frac{\phi(t)}{t - z}dt = 0 $ $ \forall z \notin \overline D $ if ...
7
votes
1answer
169 views

Evaluation of $\int_{0}^{1}\frac{\ln x}{x^2-x-1}dx$

Evaluation of $\displaystyle \int_{0}^{1}\frac{\ln x}{x^2-x-1}dx$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\infty}\frac{\ln ...
4
votes
2answers
85 views

Evaluation of Definite Integral

Evaluation of $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\cos x\sin 2x \sin 3x}{x}dx$ $\bf{My\;Try::}$ Let $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\cos x\sin 2x \sin 3x}{x}dx = ...
3
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1answer
92 views

Looking for a closed form of $I(n,m)=\int_0^{+\infty} e^{-ax^n-\frac{b}{x^m}} \, dx$

I am looking for a closed form of this integral for reals $a,b>0$ and integers $n,m>0$ $I(n,m)=\int_0^{+\infty} e^{-ax^n-\frac{b}{x^m}}$ I read about $I(2,2)$ in the book Irresistible ...
0
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0answers
28 views

Finding the area of a circle that is formed by cutting a sphere.

Say I have a sphere $x^2+y^2+z^2=a^2$ and a plane $x+y+z=b.$ How do I find the surface area of this surface? I think I would use the surface integral and for graphs the surface "element" is ...
0
votes
1answer
45 views

Gauss-Legendre three point rule

Use the change of variables $$x=\frac{a+b}{2}+\frac{b-a}{2}t,$$ to show that $$\int^b_a f(x) \ dx = \frac{b-a}{2} \int^1_{-1} f\left( \frac{a+b}{2} + \frac{b-a}{2}t \right) \ dt.\tag{1}$$Hence ...
1
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0answers
27 views

finding the Lebesgue measure of the set $A = \{ (x,y,z) \in R^3 | y^2 + z^2 \le 4, 0 \le x \le 4, x \le 6-y-z \}$.

I want to find the Lebesgue measure, denoted $u$, of this (closed and thus measurable) set. $$A = \{ (x,y,z) \in R^3 | y^2 + z^2 \le 4, 0 \le x \le 4, x \le 6-y-z \}$$ This is a cylinder cut by a ...
2
votes
3answers
32 views

Integral conversion to polar coordinates - bounds

I have an integral $$\int_0^1 \int_0^1\sqrt{x^2+y^2}\ dxdy $$ and its result is $\approx0.765...$ I convert it to polar coordinates and get $$\int_a^b \int_c^dr\ drd\phi $$ But how can i compute ...
2
votes
1answer
42 views

how could we compute this infinite real integral using complex methods?

$\int^{\infty}_{-\infty} \frac{cos(x)}{x^4+1}dx$ I know a similar result, but I'm not sure if I can take it for granted, that $\int^{\infty}_{-\infty} \frac{cos(x)}{x^2+1}dx = \frac{\pi}{e}$ The ...
0
votes
1answer
37 views

Proving integration formula involving the form a+bx

While trying to memorize and understand various integration formulas, I came across an integration rule stating that $$ \int \frac{1}{x^2(a+bx)^2} dx = ...
2
votes
4answers
90 views

Prove if $f:[a,b]\rightarrow \mathbb{R}$ is continous and not everywhere zero, then $\int_{a}^{b}f^2(x)dx>0.$

Claim: If $f:[a,b]\rightarrow \mathbb{R}$ is continuous and not everywhere zero, then $\int_{a}^{b}f^2(x)dx>0.$ I have the following theorem that seems applicable here - more specifically the ...
2
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0answers
33 views

Spherical, polar coordinates, volume of set.

Find the volume: $$\{(x,y,z)\mid x^2+y^2 \leq (z-1)^2 \leq 4-\frac{x^2}{2} - 2y^2, z\geq 1 \}$$ I've got the intersection of the following two basically: \begin{align} 1. & & & (z-1)^2 ...
2
votes
4answers
110 views

$\int_{-a}^{0}f(x) dx=\int_{0}^{a}f(x) dx$ for an even $f$

$f:\mathbb R \to \mathbb R$ integrable (but not necessarily continuous) on $\mathbb R$ function such that $f(x)=f(-x) , \forall x \in \mathbb R$. I need to show that $\int_{-a}^{0}f(x) ...
1
vote
1answer
48 views

Error for Trapezoidal Rule in multi-variable integrals

For one dimension integrals $\int_{a}^{b}f(x)dx $, we know the global truncation error goes like$\ \approx\mathcal{O}(h^2)$ where $h=\frac{b-a}{N}$ and N is the number of intervals. Also knowing how ...
3
votes
0answers
26 views

Some sort of generalized Jensen inequality?

Let $(X, \mathcal{A},\mu)$ a measure space such that $\mu(X) > 0$ and let $f, g : X \rightarrow (0,\infty)$ be such that $f, g, f\log(f), f\log(g) \in L^1(\mu)$. Show that $$ \|f\|_1\log ...
0
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1answer
37 views

Solving a differential equation with only one variable

So for some reason I'm blanking on solving a seemingly easy question: $(8x + \cos(x))\mathrm dx + (4x^2 + 7\sin(x) + 1)\mathrm dy = 0$ Which I'm currently assuming is going to be solved using an ...
2
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0answers
35 views

Solving a 1D integral with system of equations for retarded electromagnetic fields

I need to solve the following integral to calculate the effect of retarded electromagnetic fields on a test charge: ...
1
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4answers
126 views

Error in solving $\int \sqrt{1 + e^x} dx$ .

I want to solve this integral for $1 + e^x \ge 0$ $$\int \sqrt{1 + e^x} dx$$ I start by parts $$\int \sqrt{1 + e^x} dx = x\sqrt{1 + e^x} - \int x \frac{e^x}{2\sqrt{1 + e^x}} dx $$ Substitute ...
0
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1answer
36 views

Is the Riemann integral defined with partitions with subintervals of the same legth different from the general case?

When considering Riemann sums we partition the closed interval $[a,b]$ in subintervals that don't necessarily have to have the same length. Then the Riemann integral is defined by taking the supremum ...
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1answer
36 views

Show that $\int_{\tau}^{\infty}\frac{\exp(u)}{[1+\exp(u)]^2}du=\frac{\exp(-\tau)}{1+\exp(-\tau)}$

How $$\int_{\tau}^{\infty}\frac{\exp(u)}{[1+\exp(u)]^2}du=\frac{\exp(-\tau)}{1+\exp(-\tau)}$$? My Attempt : let $z=1+\exp(u)$ $\frac{dz}{du}=\exp(u)$ $$ \begin{array}{c|c|c|} ...
0
votes
2answers
16 views

About the interpretation of line integrals

I've been asked to compute the line integral of the function $f(x,y)=xy$ over the elipse $\frac{x^2}{4}+y^2=1$ counterclockwise orientated. My doubt is if this means that i have to compute the surface ...
0
votes
2answers
86 views

What is wrong with the argument that $\frac d{dx} \int_0^1 f(x)dx$ should always be $0$ for any $f(x)$?

What is wrong with the argument that $\frac d{dx} \int_0^1 f(x)dx$ should always be $0$ for any $f(x)$? My book used differentiation under the integral sign to evaluate an integral. The integral was ...
2
votes
1answer
53 views

Integrating a Complex Exponential Function

Suppose $w=\exp(2i\pi/3)$. How would I go about integrating $$\int\frac{3dx}{e^x+e^{wx}+e^{w^2x}}$$ Is there a transformation i can use? This is an entire function; there is no $x$ that will ...
1
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1answer
32 views

Checking whther the integral $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ convergent

I need to check convergence of $\int_1^∞ \frac{1}{x^{\frac{1}{x}+1}} dx$ . I think it divergence cause it bigger than $\int_1^∞ \frac{1}{x} dx$ but I can't prove it. I have an hint that ...
0
votes
1answer
19 views

Delta function of two variables in integral of three variables

I want to integrate a function like: $$ Z = \int\int\int dt dt' dt'' \, F(t'',t)G(t,t') \delta(t-t')$$ where $F$ and $G$ are arbitrary functions. Clearly we must have $G(t,t)$ in the expression due ...
5
votes
1answer
63 views

Lebesgue integral - no dominating integrable function of $(f_n)$

Let $\lambda$ be the Lebesgue-measure on $\Omega =[0,1]$. Given a sequence of non-negative measurable functions $$f_n:\Omega\to\Bbb R: x \mapsto ne^{-nx},$$ how can I show that $f_n$ converges ...
1
vote
2answers
31 views

Convergence of integral power of $\cos$

Find the integral and find $k$ in order to converge ($k$ is real number). $$\int_0^{\frac{\pi}{2}} \cos (\theta) ^{2k} d\theta.$$ I can find the value of integral if $k$ is integer, but what happens ...
0
votes
1answer
29 views

Integral problem, evaluating the substitution at zero…

So I have to show the following: $$\int_0^v \frac{x^a}{(x+k)^{2a+2}} dx = \int^{\infty}_{k^2/v} \frac{u^a}{(u+k)^{2a+2}}du$$ By making a suitable substitution. Where $k>0$ and $a$ is a positive ...
1
vote
0answers
29 views

Help with simple integral

Id like to know why this is wrong: $\int sen(x)·cos(x) dx$$\underset{\uparrow}{=}\int u\ du=\frac{u^2}{2}+c=\frac{1}{2}sen^2(x)+c\\\boxed{CV\\u=sen(x)\\du=cos(x)dx}$ When checking on Wolfram Alpha ...
2
votes
1answer
43 views

What is running integral? [closed]

My question might be too simple. But I could not find any source giving the answer. Can you please explain the running integral?
0
votes
2answers
29 views

Indefinite integration of a fraction with a non-factorable denominator

Solve the integral below: $$ \int \frac{x+1}{x^2-4x+6} \, dx $$ I tried u-sub and got $$ u=x^2-4x+6 $$ $$ du = 2x - 4 dx \leftrightarrow dx = \frac{1}{2(x-2)}du $$ $$ \int \frac{x+1}{u} ...