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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0answers
30 views

Is it possible to integrate vector fields directly?

On vector fields I mean $V\rightarrow V$ maps, where $(V,\langle,\rangle)$ is a finite dimensional inner product space, not in manifold context. I am confused about some (well a lot) aspects of ...
2
votes
1answer
37 views

How to solve this problem except calculate it directly

$$ \int_0^\infty 1-\left(1-e^{-2w}\right)^8 dw $$ I really don't know how to solve this problem except calculate it directly.
0
votes
0answers
12 views

Integral of function and its derivative, multi-dimension

Is the following statement true? If so, how can I prove it? $f = f(x,a)$ $\int \left(f(x,a_1)\int f(x,a_2)dx\right)dx = \frac{1}{2} \int f(x,a_1) dx \int f(x,a_2) dx$
5
votes
4answers
101 views

Integrating $\int \frac{dx}{x^2+x+1}$

I am trying to evaluate the following integral: $$I=\int \frac{dx}{x^2+x+1}$$ I am not supposed to do it with complex numbers so it's kind of hard. I checked the answer on WolframAlpha. It gives ...
2
votes
2answers
33 views

Evaluate fourier coefficient of $f(t)=t$.

Evaluate the Fourier coefficient of $f(t)=t$. $$\hat{f}(n) = \frac{1}{2\pi}\int_0^{2\pi} te^{-int}dt$$ I'd be glad for help with this calculation. My integration skills need an improvement. My ...
2
votes
1answer
38 views

Integral of the convolution of two functions: $\int_{-\infty}^{\infty} (f*g)(x)dx$

There is this proof for the integral of convolution between two functions: $$\begin{align}\int_{-\infty}^{\infty} (f*g)(x)dx&=\int_{-\infty}^{\infty}\left [ ...
1
vote
1answer
25 views

Double integral for gamma function

I have the following double integral $$\int_0^{2\pi}\int_0^\infty r^{p+q-1}\cos^{p-1}({\theta})\sin^{q-1}({\theta})e^{-r(\sin{\theta}{\cos{\theta}})}drd\theta $$ However, I can't seem to be able to ...
4
votes
5answers
208 views

Integral evaluation (step-by-step)

I'm trying to evaluate the integral by exponent. Could you help me with following steps? Integral: $$\int \frac{1}{4+\sin(x)} dx$$ $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$ $$\int \frac{1}{4+sin(x)} dx ...
0
votes
1answer
11 views

Transform integral bounds for multidimensional integral

I need to transform the followin integral borders into something I can use to integrate the argument of $$\int_{R^N, \; \|\vec{x}\| \lt \gamma} f(\|\vec{x}\|) d\vec{x}$$ analytically. The ...
0
votes
1answer
24 views

Using $e^{ix}$ instead of sine and cosine in contour integration

A while ago I asked: Evaluation of $\int_{0}^{\infty} cos(x)/(x^2+1)$ using complex analysis. Instead of using $\cos(z)$ an answerer said that is valid to use $e^{ix}$ How is this valid? I dont ...
4
votes
5answers
92 views

$f(0)=0$ and $\lvert\,f^\prime (x)\rvert\leq K\lvert\,f(x)\rvert,$ imply that $f\equiv 0$.

I'll start with the precise statement of the problem: Suppose that $f:[0,b]\to\mathbb{R}$ is differentiable, $\,f(0)=0$, and that there exists a real number $K\geq 0$ such that ...
0
votes
0answers
37 views

Check if a function is L2

I want to check if a function $f$ defined on $[0,T]$ is a $L_2$ function. What I know is $f$ is a $L_1$ function. (but $f$ could be not bounded) So I want to use an inequality like $$ ...
3
votes
0answers
46 views

Evaluation of $\int_{-\infty}^{\infty}\operatorname{e}^{-\mu x^2}f(\nu x)\operatorname{d\!}x$ for $\mu>0$

I'm trying to evaluate the integral $$ \Psi(\mu,\nu)=\int_{-\infty}^{\infty}\operatorname{e}^{-\mu x^2}f(\nu x)\operatorname{d}\!x\qquad(\text{for}\; \mu>0)\tag 1 $$ where $\nu\in\Bbb R$ and $f$ ...
1
vote
2answers
53 views

$\int_0^1e^{(ax^2 + bx)}dx$ in terms of erf

I'm trying to evaluate the definite integral, $$\int_0^1e^{(ax^2 + bx)}dx$$ in terms of the function, $$F(z)=\int_0^ze^{p^2}dp$$ The correct answer that i'm supposed to get is, ...
5
votes
3answers
71 views

Evaluation of $\int_{0}^{\infty} cos(x)/(x^2+1)$ using complex analysis.

Evaluate: $$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ Using only complex analysis. $$I = \int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = (\frac{1}{2})\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 ...
2
votes
0answers
23 views

Choosing a contour to integrate over.

What are the guidelines for choosing a contour? For example to integrate a real function with a singularity somewhere. What type of contour from Square, keyhole, circle, etc should be chosen for ...
0
votes
1answer
55 views

How to check if functions are integrable?

Consider two functions $$ \int_0^1 \frac{1}{e^x-1} dx $$ and $$ \int_0^1 \frac{1}{(e^x-1)^2} dx $$ How to check if these functions are integrable?
4
votes
3answers
186 views

Integration without complex analysis on rational-improper integral

Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} \,\mathrm dx$$ Without the use of complex-analysis. With complex analysis it is a very simple problem, how can this be done WITHOUT complex analysis? ...
0
votes
1answer
43 views

Proving integration techniques with intuition.

I've recently competed my A levels and now that I'm in the university I finally found the time to understand calculus on a intuitive level. So I've been reading up on books such as "Calculus with ...
3
votes
2answers
104 views

Definite integral $\int_0^{2\pi}\frac{1}{\cos^2(x)}dx$

I encountered this very simple problem recently, but I got stuck on it because I think I am missing something. It is easy to see that indefinite integral $\int\frac{1}{\cos^2(x)}dx$ is $\tan(x)+C$. ...
2
votes
1answer
180 views

How to do contour integral on a REAL function?

Suppose we are given the problem: Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} dx$$ Where $x$ is a real variable. A real variable function (no complex variables). I was reading Schaum's ...
0
votes
0answers
28 views

Integral problem with perfect square in denominator, $\int_{-\infty}^\infty \frac{1}{(1-\beta z^{-1})^2}\,dz$

I am trying to solve this problem, but I failed to solve it several times. It is very difficult for me. $$\int_{-\infty}^\infty \frac{1}{(1-\beta z^{-1})^2}\,dz$$
1
vote
0answers
25 views

Switch integral and sum for Bessel function.

I haven't real knowledge in Bessel's function and I'd like to know how to switch integral and sum in these two equations. I've already tried a lot of ideas but nothing really works. The first one is : ...
0
votes
1answer
23 views

Discussing the convergence of $\displaystyle\int_I\frac{x+2}{\sqrt x\left(x^2+x+1\right)^4}\mathrm dx$

Let $$f(x) = \frac{x+2}{\sqrt{x}\left(x^2 + x + 1\right)^4}$$ Discuss the convergence of $\displaystyle\int_0^1f(x)\,\mathrm dx$ and $\displaystyle\int_1^{+\infty}f(x)\,\mathrm dx$. I encountered ...
0
votes
0answers
47 views

Where did I go wrong with this definite integration?

I'm trying to solve the definite integral $\int_0^n\pi^{ex}dx$ Wolfram says that the answer is $\frac{\pi^{en}-1}{e \ln(\pi)}$, but I got $\frac{\pi^{en}-1}{\ln(\pi)}$. Can anyone help me figure out ...
0
votes
0answers
60 views

Nice approximations of sums by integrals.

Let $f(x):\Bbb Z^+\rightarrow \Bbb R^+$ be a non-monotone function. If for every $m\in\Bbb N$, $$S(m) =\sum_{n=1}^N\frac{1}{(1+f(n))^m}$$ be sum of interest, then is there a way to approximate this ...
1
vote
1answer
39 views

$L^1$ and $L^2$ norm inequaliy

Consider real valued function $f$ defined on $[0, T]$. L1 norm and L2 norm of function $f$ are given by $$ \|f\|_1=\int_0^T |f(t)| \, dt $$ and $$ \|f\|_2=\sqrt{\int_0^T |f(t)|^2 \, dt } $$ Then we ...
0
votes
0answers
27 views

Movement of Horse Position during a race

I am trying to determine how to trace a horses position in running during a race and sort them in order of the horses have the fastest foot speed. Here is a sample of the data: ...
-1
votes
0answers
35 views

Check computation of conditional covariance

Note: HERE YOU CAN SEE THIS PAGE. Explanation of an integral formula for the expectation of $(X_1-X_2)(Y_1-Y_2)$
3
votes
2answers
92 views

difficult complex integral $\int_\gamma \frac{1}{z^2+i}dz$

We are asked to calculate $\int_\gamma \frac{1}{z^2+i}dz$ where $\gamma$ is the straight line from $i$ to $-i$ in that direction. My parametrization is simple, I chose $z(t)=i-2it$. Notice that ...
5
votes
1answer
112 views

Evaluate $\int \ln(1 + e^x)\ \mathrm dx$

Evaluate the following indefinite integral. $$\int\ln(1 + e^x) \mathrm dx$$ My attempt :: Using integration by-parts, \begin{align} \int\ln(1 + e^x)\cdot 1\ \mathrm dx &= x\ln(1 + e^x) - \int ...
3
votes
2answers
39 views

Convergence of a integral: $\int_{0}^{1} |\ln (x)|^n \ dx$

Let $n \in \mathbb N$ be arbitrary. Does the integral $$\int_{0}^{1} |\ln (x)|^n \, dx$$ converge? I asked myself this question and I have no idea of a proof or counter example. Someone can give me a ...
3
votes
0answers
63 views

Evaluate Integral [duplicate]

Find $\displaystyle\int_0^\infty\frac{\sin^4x}{x^4}$ using the fact that $\displaystyle\int_0^\infty\frac{\sin^2x}{x^2} = \frac{\pi}{2}$. The graph of $\dfrac{\sin^4x}{x^4}$ was also given, I tried to ...
0
votes
0answers
14 views

Evaluate the given integral along the given (positively oriented) circle. [on hold]

Ok, so I have the following problems that I am working on. It says to evaluate 1) where C is given by |z+1|=1/2 2) where C is given by |z-2|=1/2 3) where C is given by |z|=2 4) where C ...
-1
votes
0answers
51 views

Find $ \int_{\theta_0}^{\theta} \cos \theta \left( \sin 2\theta \right)^{3/2} \, \mathrm{d}\theta $ [on hold]

Find $$ \displaystyle\int_{\theta_0}^{\theta} \cos \phi \left( \sin 2\phi \right)^{3/2} \, \mathrm{d}\phi $$
1
vote
4answers
72 views

The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$

Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is ...
-2
votes
1answer
41 views

Evaluating a complex integral using the Cauchy integral formula [on hold]

I need to evaluate the following integral counterclockwise: $$\oint_{\left | z \right |=\frac{1}{2}} \frac{dz}{(z-1)\sin z} $$ using the Cauchy integral formula
3
votes
2answers
31 views

Laplace transform of $f(t)=te^{-t}\sin(2t)$

I was asked to find the laplace transform of the function $f(t)=te^{-t}\sin(2t)$ using only the properties of laplace transform, meaning, use clever tricks and the table shown at ...
2
votes
0answers
14 views

Bounding $\int_{\infty}^{\infty}|g(s)v^3k(v)|dv$ where $k$ is a second-order kernel

Suppose $k$ is a nonnegative, bounded real-valued function that satisfies $$ \int_{-\infty}^\infty k(v)dv=1,\quad k(v)=k(-v),\quad \int_{-\infty}^\infty ...
1
vote
0answers
34 views

Joint and marginal distributions and expectations (Is my proof right?)

1. Please look this following proof first: 2. I want to proof the conditional case, and the proof process is following. 3. I want somebody to help to check whether my proof is right? Thanks
5
votes
2answers
63 views

Quadratic Expressions: Advanced techniques of Integration

$$\int \frac{x}{\sqrt{5+12x-9x^2}}\,dx$$ After two steps I arrive at $\displaystyle{ \int \frac{x}{\sqrt{9-(3x-2)^2}}}\,dx$ Using trigonometric substitution, we have a triangle with a cosine of ...
0
votes
2answers
37 views

Continuous piecewise smooth curve

I cannot understand the definition of $\tilde d(p_1,p_2)$ here? Can anyone please explain it clearly?
1
vote
1answer
42 views

Advanced Integration techniques: Quadratic Expressions and U-Substitution

Find $$\int \frac{2x-1}{x^2-6x+13}dx $$ In the final steps after a u-substitution, one arrives at $$\int \frac{2u}{u^2+4}du + \int\frac{5}{ u^2+4}du$$ The next step is arriving at $$\ln(u^2+4) + ...
1
vote
1answer
26 views

Finding pathline

I've been trying to find the pathline of a particle dropped in a steady flow defined by the following vector components: $$ u= \frac{-2x}{(x^2+y^2+1)^2} \hat i + \frac{-2y}{(x^2+y^2+1)^2}\hat j $$ in ...
1
vote
2answers
52 views

Solutions to the integral $\int \frac {dx}{2\sqrt x (x+1)}$

I am given a question to solve the integral $\int \frac {dx}{2\sqrt x (x+1)}$. When I substitute $x+1 = t^2$, I get the solution as $\space \ln(\sqrt{x+1} + \sqrt x) +C$; while when I substitute ...
-3
votes
1answer
76 views

I do not understand the last step of this proof. [on hold]

1. PLEASE LOOK THE FOLLOWING PROOF FIRST. 2. Suzu explained the fist several steps to me in this page :Explanation of an integral formula for the expectation of $(X_1-X_2)(Y_1-Y_2)$ . But I still ...
0
votes
2answers
33 views

Evaluation of an integral of some expressions involving fractions

I am stuck in evaluating the following integral: \begin{equation} \int_{0}^{b-a} \frac{1}{\sqrt{u} (a+u)} \,du, \end{equation} where $0<a<b$. Any ideas?
1
vote
1answer
43 views

An integral with density function of $N(\hat{a}, \frac{1}{s})$

I am stucked on this integral, which is from a research paper in Finance, for a while, so can anyone please help walk me through how we can get the answer on the RHS of this integral? Prove: ...
2
votes
0answers
54 views
+150

A treatise on Probabilistic arguments and Laplace/Fourier transforms to solve limits/integrals from basic calculus.

I've seen in some answers in Brilliant.org to some very complicated limits and integrals that uses probabilistic arguments (Let $X$ be a random variable from $[0,1]$... some examples are in those ...
5
votes
4answers
168 views

Finding $ \int_0^1 \frac {\ln x}{1+x^2}\mathrm dx $

Today I encountered the problem of how to find $$ \displaystyle\int_{0}^{1} \frac {\ln x}{1 + x^2}\mathrm dx $$ but got no start on it. Is this one of those integrals which we have to approach from ...