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 ...

learn more… | top users | synonyms (3)

1
vote
1answer
55 views

Indefinite integrals and changing variables

Why is it legal to change the variable of an indefinite integral? Consider $$\int \dfrac{dx}{\cos x}$$ If one were to say, $\text{Let } u=\cos x$, do we not now technically have ...
0
votes
0answers
50 views

First variation of convolution of two nonlinear functions, how to reexpress $\left[x \delta x * x^2 \right]$?

A new variational principle is presented in this paper: Mixed Convolved Action When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set ...
2
votes
1answer
97 views

Integration in PID controller

I am trying to understand how come there is a phase difference is from the error signal and the output of my PID controller which consisting of I = 1. As far i've understood should the integration ...
3
votes
2answers
122 views

Evaluate the definite integral $ \int_{-\infty}^{\infty} \frac{\cos(x)}{x^4 +1} \ \ dx $

I am having more trouble with this problem then I feel like I should be. I set $ \ \cos(x) = e^{ix} \ $ and $ \ x^4 +1 = e^{\pi i /4} \ $ or $ \sqrt{i} \ $. I think I am suppose to do a residue to ...
2
votes
0answers
33 views

Is little-o preserved under integration and derivation of another variable?

Given an integrable function $g:\mathbb{R}\longrightarrow\mathbb{R}$, and a function $f:\mathbb{R}^2\longrightarrow\mathbb{R}$ such that $f(x,y)=o(x^{-1})$ when $x\rightarrow\infty$, i.e. ...
-1
votes
1answer
49 views

Strange Integral Notation? [duplicate]

When reading about an certain algorithm (about parameter estimation for Kalman Filtering page 7 eq 57) I found this notation: $\int dx f(x)$ which is normally written as $\int f(x) dx$. I spent a ...
1
vote
4answers
62 views

How to calculate this limit involving an integral

I missed a couple of classes so I'm having trouble doing this (and other similar) excercises of homework: $\lim\limits_{x\to0^+}\frac{\displaystyle\sqrt{x}-\displaystyle\int_0^\sqrt{x} ...
3
votes
2answers
76 views

Trigonometric integral with the function- sin

I need some help with this integral please: $\displaystyle\int x\sin\frac{1}{x}dx$
1
vote
2answers
63 views

Area calculation of ellipse $x^2/2+y^2=1$

Calculate the area of the ellipse that you get when you rotate the ellipse $$\frac{x^2}{2}+y^2= 1$$ around the x-axis. My approach has been to use the formula for rotation area from $-2$ to $2$. But ...
3
votes
1answer
67 views

$\int_0^1 e^{x^2}(2x-a)dx=0$

$\int_0^1 e^{x^2}(2x-a)dx=0$ where $a$ is any real number then predict the range of $a$ or find its value. My approach: $\int_0^1 e^{x^2}2xdx=(e-1)$ Can't integrate $a\int_0^1 e^{x^2}dx$.
8
votes
6answers
318 views

Calculus Question: $\int_0^{\frac{\pi}{2}}\tan (x)\log(\sin x)dx$

Can anyone help me to find $\int_0^{\frac{\pi}{2}}\tan (x)\log(\sin x)dx$? Any help would be appreciated. Thanks in advance.
6
votes
3answers
65 views

Evaluating integral with $\sqrt[3] x$ and $\ln x$

I found this example amongst set of examples to get ready on exam $$ \int\frac{\ln x}{\large\sqrt[3]{x}}\ dx $$ I am able to see the basic substitution, but I don't know how to count if further… ...
4
votes
2answers
48 views

Evaluating $\int \sqrt{x}\ln(2x)\:\mathrm{d}x$

I am learning to an exam and stumbled upon this integral. $$\int \sqrt{x}\ln(2x)\:\mathrm{d}x$$ the result should be $\frac{2}{3}\sqrt{x^{3}}\left(\ln(2x)-\frac{2}{3}\right)+C$, but I am still ...
0
votes
1answer
50 views

I stuck with this integral (lnx)

Good morning, I would like to ask you guys with helping me out with this integral, still can't get to the proper result...
3
votes
4answers
46 views

Per partes integration

Can I asked you guys for help with solving of this integral? Thank you. It could go well with the per-parted method, but I am not able to finish it. $$\int x^2 \ln x dx$$
1
vote
0answers
58 views

[Measure theory]. Proof of inequality of integrals of simple function

I have a question regarding a proof in my textbook. The theorem is as follows : if {$f_n$} is an increasing sequence of non-negative simple functions and $lim_{n \rightarrow \infty}f_n(x) \geq g(x), ...
8
votes
2answers
116 views

Evaluate $\displaystyle\int_0^\pi \frac{x}{1+\sin^2x} \ dx$

How can one evaluate $$\int_0^\pi \frac{x}{1+\sin^2x} \ dx\ ?$$
1
vote
1answer
50 views

Complex integration where the limits are complex numbers

I've been reading about integration in $\Bbb C$, and things look pretty similar to multivariable integration, however I found a series of excersices that baffled me a little, I don't know how to solve ...
0
votes
1answer
32 views

Find the volume of a cone whose length of its side is $R$

How can i compute the volume of a cone whose length of its side is $R$ and the vertex of the cone forms an angle $2θ$ . The top cone is a cap of a sphere of radius $R$. I tried to solve first in 2 ...
1
vote
0answers
34 views

Evaluating Surface Integrals

I've been told that surface integrals can be evaluated through projection by $$\iint_S f(x,y) \, dS = \iint_R \frac{1}{\| \frac{\mathbf{n}}{\|\mathbf{n}\|} \cdot \mathbf k \| }f(x,y) \,dx\,dy$$ ...
9
votes
4answers
201 views

Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$ using complex analysis.

Evaluate $\int_0^\infty \frac{(\ln x)^2}{x^2+4} \ dx$. This is the last question in our review for complex analysis. Hints were available upon request, but being the student I am, I waited until the ...
0
votes
2answers
100 views

Trouble with Indefinite Integral - $\int \frac{2x^{3}+8x}{\sqrt{(x^{2}+4)}}dx$

$$\int \frac{2x^{3}+8x}{\sqrt{(x^{2}+4)}}dx$$ I'm having issues integrating this. I found a step through and I do not understand the method of substitution. Would anyone be able to give me a brief ...
1
vote
1answer
26 views

Substitution integral with sines and cosines

Could I ask you guys to help me out with this? $\displaystyle\int \frac{\sin x}{\sqrt{2+\cos x}}dx$ Thank you in advance
1
vote
0answers
49 views

Finding reasonable change of variables?

Let $R \subseteq R^{2}$ be the first quadrant region bounded by the curves $x^2 + y^2 =4, x^2 + y^2 =9, x^2 −y^2 =1$ and $x^2 − y^2 =4$. Use an appropriate change of variables to evaluate. ...
1
vote
2answers
60 views

$\int \text{sech}^4 x \, dx$

How we can solve this? $$ \int \text{sech}^4 x \, dx. $$ I know we can solve the simple case $$ \int \text{sech} \, dx=\int\frac{dx}{\cosh x}=\int\frac{dx\cosh x}{\cosh ^2x}=\int\frac{d(\sinh ...
2
votes
5answers
74 views

Integration with Euler number and sin2x

I found this example in a textbook: $$\int e^{\cos^2 x}\sin2x dx$$ There are also results, but I am not even close to that...
0
votes
2answers
33 views

Integral with goniometric functions

I am learning on an exam and found this example: $\int cos^3x sinx dx =$ How do determine it here? Do I need to apply here some goniometric rules? And with substitution? Thank you
1
vote
2answers
69 views

Find the area of the surface of revolution generated by revolving about the $x$-axis the hypocycloid $x=a\cos^3\theta$, $y=a\sin^3\theta$

Find the area of the surface of revolution generated by revolving about the $x$-axis the hypocycloid $x=a\cos^3\theta$, $y=a\sin^3\theta$ ($0 \leq \theta \leq \pi$) I know you have to integrate $2\pi ...
2
votes
2answers
54 views

How to finish this integration?

I'm working with the integral below, but not sure how to finish it... $$\int \frac{3x^3}{\sqrt[3]{x^4+1}}\,dx = \int \frac{3x^3}{\sqrt[3]{A}}\cdot \frac{dA}{4x^3} = \frac{3}{4} \int ...
2
votes
1answer
44 views

Substitution on integral

I have this (it looks simple) example $$\int \frac{3}{2-5x}\,dx$$ It looks really simple, no logarithms or trigonometric functions, but I just cannot get the proper result here...
2
votes
1answer
59 views

Reduction formula integral of $\cos^n(x)$ [duplicate]

I need to find the reduction formula for the integral of $\cos^n(x)$. Ive split it into $\cos(x)\cos^{(n-1)}x$ in the hope of integrating by parts, but I'm unsure how to differentiate $\cos^{(n-1)}$, ...
12
votes
5answers
366 views

Integral $\int_0^\infty \log^2 x\frac{1+x^2}{1+x^4}dx=\frac{3 \pi^3}{16\sqrt 2}$

This integral below $$ I:=\int_0^\infty \log^2 x\frac{1+x^2}{1+x^4}dx=\frac{3 \pi^3}{16 \sqrt 2} $$ is what I am trying to prove. Thanks. We can not expand the denominator as a series since the ...
2
votes
1answer
44 views

integrals of vector fields that yield vectors, not scalars

When I tried to think of how I'd answer this question, I realized that never in my undergraduate curriculum was I asked to compute the surface or line integral of a vector field. I don't mean I've ...
1
vote
1answer
78 views

Integral/Vector calculus $\oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S.$

I am trying to show that $$ \oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S $$ using Levi Cevita notation methods only. The Levi ...
0
votes
1answer
39 views

Given a $H(x)$ definite integral of $ f(t)$, determine $f$ and the constant knowing that is continuous

Considering the function $$H(x)= \int_{0}^{\pi\cdot x}f(t)dt=\frac{(2 \cdot \sin(\pi\cdot x)-\cos(2\pi\cdot x)+2\pi^2t+k)}{ 2}$$ determine $f$ and $k$ knowing that $f$ is continuous on ...
1
vote
1answer
21 views

Integral determination

I am trying to figure out this integral: $\int \frac{x}{(x^2+4)^6}dx$ Substitution: $t = x^2+4$ $dt = 2xdx => dx=\frac{dt}{2x}$ Then: $\int \frac{x}{(x^2+4)^6}dx = \int ...
0
votes
1answer
34 views

Find f with A plane curve whose equation is $y - f (x) = 0$ passes through the origin.

A plane curve whose equation is $y - f (x) = 0$ passes through the origin.Consider the rectangle $R_x$ formed by the coordinate axes and lines parallel to the axis passing through the point $(x, f ...
2
votes
4answers
145 views

Integral $\int_0^1 \log x \frac{(1+x^2)x^{c-2}}{1-x^{2c}}dx=-\left(\frac{\pi}{2c}\right)^2\sec ^2 \frac{\pi}{2c}$

Hi I am trying to prove this result $$ I:=\int_0^1 \log x \frac{(1+x^2)x^{c-2}}{1-x^{2c}}dx=-\left(\frac{\pi}{2c}\right)^2\sec ^2 \frac{\pi}{2c},\quad c>1. $$ Thanks. Since $x\in[0,1] $ we can ...
1
vote
1answer
145 views

Solving Volterra integral equation of first kind with a Gaussian diffusive evolution kernel

I am trying to solve following Voltera integral equation for $P(t|t')$ numerically: $$ \rho(1,t|0,t') = \int_{t'}^{t} dt'' \rho(1,t|1,t'') P(t''|t') $$ where $$ \rho(x,t|x',t') = ...
6
votes
1answer
157 views

Measure theory questions applied to Second Order PDE

Most of the questions are more measure theory and integration related but I need to give some context, so I will now. Consider the quasilinear 2nd-order partial differential equation ...
2
votes
1answer
62 views

Do simple functions converge almost everywhere?

Assume there is a sequence of simple functions s.t.: $$\|\int(s_m - s_n)\mathrm{d}\mu\|\to 0$$ Does it follow that there is a subsequence which converges almost everywhere? (Note the order of modulus ...
0
votes
2answers
36 views

Given a $M(x)$= definite integral of $ f(t)dt$, determine$ f$ and the constant knowing that is continuous

Considering the function $$M(x)= \int_{4}^{x^2}f(t)dt=e^{2x^2}+ x^4+8x^2+k$$ determine f and k knowing that f is continuous on ℝ. My steps(not sure if are right) For get f: derivate $e^{2x^2}+ ...
0
votes
1answer
36 views

measure theory problem

I am stuck at part d) of this problem. Do you see how to show that f is measurable? I must show that $f^{-1}[-\infty,r)$ is measurable for all r. I am not sure how to do it. I would assume that it ...
2
votes
1answer
212 views

(High school) How to become an expert on integration?

It is very hard for me sometimes to find a right method to do an integration, for example, whether to use by parts method or substitution. I always find the questions I was given on an exam very hard ...
1
vote
0answers
50 views

Repository of functions which do not have elementary integrals [duplicate]

If there is some function and I suspect that the primitive function cannot be expressed using elementary functions, I would like to have some argument that there indeed is no such expression. One ...
0
votes
1answer
30 views

$f$ is continuous and satisfies the equality given for all $0 \leq x $

How can i compute $f (2)$ if $f$ is continuous and satisfies the equality given for all $0 \leq x $ : $\int_{0}^{f(x)} t^2 dt = x^2(1-x)$ and $\int_{0}^{x^2(1-x)} f(t) dt = x$ Some help for this ...
1
vote
1answer
46 views

Length of a polar curve, $r= 2\theta^2$

$$r = 2\theta^2, 0<=\theta <=\sqrt{5}$$ calc the length of the curve. Since it's probably polar coordinats the formulat should be: $$\int_0^\sqrt{5} \sqrt{(r(\theta))^2+(r'(\theta))^2} d\theta ...
1
vote
2answers
59 views

integrating exponential

How can you integrate $\frac{e^{x-3}}{x^4} dx$. I tried integration by parts but it's not possible. Is some substitution possible ? I started off solving the diff eqn $(xy^2 + 3e^{x-3})dx - x^2ydy ...
8
votes
4answers
285 views

Double Integral $\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\cos(x+y)\,dx\,dy=(\gamma+2\log 2)\pi^2$

Hi I am trying to solve this double integral $$ I:=\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\cos(x+y)\,dx\,dy=(\gamma+2\log 2)\pi^2. $$ Thank you. The constant in the result is ...
2
votes
0answers
60 views

Multivariable Integral Calculus help

I have two questions. First: Is my proof "strong" enough? I am being asked to prove that $$\int_{0}^\infty\int_{0}^x e^{-sx}f(x-y,y) dydx = \int_{0}^\infty\int_{0}^\infty e^{-s(u+v)}f(u,v) dudv$$ ...