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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1answer
104 views

How to solve mechanics problem when acceleration depends on position.

I'm curious about how problems such as the following are typically solved analytically, or in computer simulations such as games engines for 2D physics. It seems a bit harder than the typical constant ...
2
votes
1answer
63 views

asymptotic approximation when $a\to 0^+$ of $I(a):=\int_0^\infty \int_0^{a/x}e^{-x-y}\ dy\ dx.$

I want to find an asymptotic approximation when $a\to 0^+$ for the integral $$I(a):=\int_0^\infty \int_0^{a/x}e^{-x-y}\ dy\ dx.$$ I found the following approximation: $$C_1\, a\, \mathrm{ln}(1/a) ...
2
votes
3answers
260 views

Distribution of stochastic integral

Assume that $\mathrm{d}S = \sigma \, \mathrm{d}W$ with initial level $S(0)$ and where $\mathrm{d}W$ is usual Brownian motion. Now $A(T) = \frac{1}{T} \int_0^T S(t) \, \mathrm{d}t$ . What is the ...
9
votes
1answer
219 views

Help in evaluating $\int \frac {t^4 \tan t}{2 + \cos t}~dt$

Can anybody help me in evaluating this indefinite integral? I can't possibly find a workable substitution: $$\int \dfrac {t^4 \tan t}{2 + \cos t}dt$$
11
votes
1answer
132 views

Maximize $\int_0^1 x^2f(x)~\mathrm dx - \int_0^1 xf(x)^2~\mathrm dx$ among continuous $f:[0,1]\to\Bbb R$

For a function $f$, let $$ a = \int_{0}^{1} x^2f(x) \mathrm{d}x\\ b = \int_{0}^{1} xf^2(x) \mathrm{d}x, $$ where $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$. Then find ...
1
vote
0answers
66 views

How to compute the arc length of $f(x) = ax + b \sin(x)$.

I would like to compute the length of the arc of $f(x) = a x + b \sin(x)$ (let's say from $0$ to $\alpha < 2\pi$.). The traditional method of computing it as the integral $\int_0^\alpha ...
0
votes
1answer
151 views

Evaluating $\int \frac {\sin x}{1+x^2}\,dx$

How do you integrate this? Tried different substitution already but I failed: $$\int \dfrac {\sin x}{1+x^2}~dx$$
0
votes
1answer
57 views

Evaluate $\int_0^4 |\sqrt{x} - 1|~dx$

I am working on a problem set involving indefinite integrals. Currently stuck in this question: $\int_0^4 |\sqrt{x} - 1|~dx$ I tried the following substitution: Let $u=\sqrt{x}$ so, ...
0
votes
0answers
87 views

Integral of exponential function with power

I would like to know if somebody has an insight about how to evaluate these 2 integrals: $$ F(s) = \int_0^\infty e^{-\alpha x^\beta-sx}dx $$ $$ I = \int_0^\infty \int_0^\infty e^{-\alpha (x+t) ^\beta ...
0
votes
2answers
35 views

$\iint_V |y-x^{2}| \operatorname{d}x \operatorname{d}y$ with $V = [-1,1] \times [0,2]$

it's especially difficult because i don't understand how to integrate absolute value terms. I only know that if you function, say $x^{2}-1$, is below the $x$-axis i need to integrate $1-x^2$ between ...
1
vote
1answer
49 views

How to calculate the inverse of the line integeral.

Let $f$ be a polynomial function, $$ f(x) = a_0 + a_1 x + ... + a_d x^d $$ where $a_0$, $a_1$, ..., $a_d$ are parameters and usually $d \le 6$. Let $g$ be the line integral of $f$, $$ g(x) = ...
-1
votes
1answer
142 views

How to solve gamma function integral for 4!

I have been trying to test my knowledge of the gamma function by calculating 1! and 4!. I got the right result for 1! but I cannot get 4! analytically. To be more specific, I think I am not ...
10
votes
3answers
318 views

Evaluating integral $\int_0^{\frac{\pi}{2}}\log\left(\frac{1+a\cos(x)}{1-a\cos(x)}\right)\frac{1}{\cos(x)}dx$

How can I evaluate following integral? $$ \int_0^{\pi/2} \log\left(\frac{1 + a\cos\left(x\right)}{1 - a\cos\left(x\right)}\right)\, \frac{1}{\cos\left(x\right)}\,{\rm d}x\,, ...
5
votes
1answer
186 views

closed form for $\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx$

How can I find a closed form for $$\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx, n\in\mathbb{N}$$
3
votes
1answer
41 views

Integration of exponential with square

It is known that $\int_\mathbb{R}e^{-tx^2}dx=\sqrt{\pi/t}$. What about $\int_\mathbb{R}e^{-t(x+ai)^2}dx$ for $a\in\mathbb{R}$? Is it still also $\sqrt{\pi/t}$? I can't simply change the variable ...
0
votes
1answer
41 views

Integral of exponential with second degree exponent

I want to compute the integral $$\int_\mathbb{R}e^{-t\left(y-\dfrac{(at+x)i}{2t}\right)^2}dy$$ I know that $\int_\mathbb{R}e^{-ty^2}dy=\sqrt{\pi/t}$, but here there is an extra imaginary factor. What ...
3
votes
1answer
52 views

Differentiation under integral sign for exponential

This question arises from this question: Suppose $P(x)$ is a polynomial. Why is it the case that $$\dfrac{d}{dy}\int_\mathbb{R}iP(x)e^{-x^2/2}e^{-ixy}dx=\int_\mathbb{R}xP(x)e^{-x^2/2}e^{-ixy}dx?$$ ...
2
votes
0answers
60 views

Numerical Methods for estimating divergence over an improper integral

Problem given a function $f(x)$, defined on $[ \epsilon, \infty )$. Is there a way to "numerically estimate" whether the integral of the function diverges over the domain $[ \epsilon, \infty )$? ...
0
votes
2answers
1k views

Use Maclaurin Series to evaluate the definite integral correct to within an error $\lt 0.0001$

Definite integral: $$\int_0^{0.2} \dfrac{1}{1+x^5}\text{d}x$$ So I did series expansion of $$\sum_{n=0}^{\infty}\dfrac{(-1)^n\cdot x^{5n}}{5n+1}+C$$ and when I plug in $0.2$ that makes it ...
1
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2answers
114 views

Integrating trigonometric function problem

\begin{eqnarray*} \int \frac{3\sin x+2\cos x}{2\sin x+3\cos x}dx &=& \int \frac{(3\sin x+2\cos x)/\cos x}{(2\sin x+3\cos x)/\cos x}dx\\ \\ &=& \int \frac{3\tan x +2}{2\tan x +3} dx\\ ...
33
votes
4answers
1k views

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}dx$

Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}dx.$$ I do not have a strong reason to be sure it exists, but I would be ...
2
votes
0answers
32 views

Riemann integrability in not sequentially complete LCS?

For $E$ any Hausdorff locally convex space, I have been wondering whether Riemann integrability of all continuous functions $f:[\,0,1\,]\to E$ implies that $E$ be sequentially complete. For example, ...
3
votes
1answer
81 views

Finding a closed form for this integral if one exists.

Find a closed form for this integral $$\int \frac{dx}{(1+x)(1+x^a)}$$ This integral has the possibility of not having a closed form in which case can it be proven? Feeble attempt so far: $$\int ...
0
votes
1answer
78 views

Integral of characteristic function is infinitely differentiable

Let $X$ be a set, $F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given a random variable $f:X\rightarrow\mathbb{R}$, define $$\chi_f(t)=\int_Xe^{itf}d\mu$$ We can ...
2
votes
1answer
220 views

Fourier transform on Hermite polynomial

Let $h_0(x)=e^{-x^2/2}$ and $h_k=B^kh_0$, where $B=-\dfrac{d}{dx}+x$. Define a transformation $T$ as $$Tf(y)=\dfrac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(x)e^{-ixy}dx$$ How can I find the ...
2
votes
0answers
15 views

Compute tetrahedral region

Show that the volume of region $A$ is $1/6$. Region $A$ is a tetrahedral region in $\mathbb R^3$. $$A=\{(x,y,z)∈R^3 \mid x\ge 0, y\ge 0, z\ge 0, \text{ and } x+y+z\le 1\}$$
1
vote
1answer
101 views

Absolute value bound of Lebesgue integral

For the Riemann integral, we have the bound $$\left|\int_Af(x)dx\right|\leq\left(\sup_{x\in A}|f(x)|\right)\cdot\left|\int_Adx\right|$$ Do we have a similar bound for the Lebesgue integral, one like ...
3
votes
3answers
840 views

How to master integration and derivation?

We have learnt in school about derivation and integration, however I find my knowledge fairly poor. I mean I have problems with taking the derivative/integrating even simple functions. So I would like ...
1
vote
1answer
41 views

Calculate the area between $f(t) = \cos(t)$ and the $t$ axis as $t$ varies from $0$ to $\pi/4$

The problem below is part of a multiple section question that I'm hoping to solve. Calculate the area between $f(t) = \cos(t)$ and the $t$ axis as $t$ varies from $0$ to $\pi/4$.
7
votes
2answers
307 views

How find this integral $I=\int_{0}^{1}\sqrt{1-W^2(x)}dx$

How find this nice integral $$I=\int_{0}^{1}\sqrt{1-W^2(x)}dx$$ where $W(x)$ is Lambert W function:see http://en.wikipedia.org/wiki/Lambert_W_function My try: let ...
5
votes
4answers
203 views

Evaluating $\int_{0}^{\pi} \frac{\cos(nx)}{(p+\cos(x))^2+q^2}\ \mathrm dx$

I have a formula in my research, but have no idea how to get the explicit formula. $$\int_{0}^{\pi} \frac{\cos(nx)}{(p+\cos(x))^2+q^2}\ \mathrm dx$$ where n is an integer.
4
votes
2answers
37 views

Integral of derivatives and conjugate

For function $f,g$ in the Schwartz class, I want to show that ...
0
votes
1answer
61 views

Prove x(t) is bounded given a integral inequality

I want to answer the following question: $x=x(t)$ is defined and continuous on $[0,T)$ and satisfies an integral inequality $$1 \leq x(t) \leq A_1 + A_2\int_0^t x(s)\big(1+\log x(s)\big) ds$$ for ...
2
votes
1answer
25 views

Integrating two variables for $L^1$ function

Suppose $f,g\in L^1(\mathbb{R})$. Is it necessarily true that $$\int_\mathbb{R}\int_\mathbb{R}|f(x-y)g(y)|dxdy=\int_\mathbb{R}|f(x)|dx\int_\mathbb{R}|g(y)|dy.$$
1
vote
2answers
62 views

Integral of $y(x)$ when $y(t)$ is in the equation

I'm supposed to find the limit of $y(x)$ when $x \rightarrow \infty$ if $y$ is given by: $$y(x)=7+\int_0^x 4\frac{(y(t))^{2}}{1+t^2}dt$$ What I don't get is the $y(t)$ inside the integral. If I ...
1
vote
1answer
167 views

Determine whether the series is convergent or divergent by expressing $S_n$ as a telescoping sum $\sum_{n=1}^{\infty}\frac{6}{n(n+3)}$

I have no idea where I'm going wrong or if I'm even doing this problem correctly. But here are my steps so far: $$\sum_{n=1}^{\infty}\frac{6}{n(n+3)}=S_n\sum_{i=1}^{n}\frac{6}{i(i+3)}$$ After ...
1
vote
2answers
57 views

Question regarding $\int \frac{e^x}{e^x-2} \,dx$

I tried to solve this integral in the following way: $$ \text{Let } u = e^x-2 \Rightarrow du = e^x \, dx \Rightarrow dx = \frac{du}{e^x} \\ \int \frac{e^x}{e^x-2} \,dx = \int \frac{e^x}{u}\, ...
0
votes
2answers
49 views

The relationship between the determinant & the integration?

I'm curious to know the relationship between the determinant and the integration by using the area of the square Let's say we have the following matrix. $$ A = \begin{bmatrix} 2 & 0 \\ 0 & 2 ...
4
votes
1answer
104 views

Understanding the solution of a telescoping sum $\sum_{n=1}^{\infty}\frac{3}{n(n+3)}$

I'm having trouble understanding infinite sequence and series as it relates to calculus, but I think I'm getting there. For the below problem: $$\sum_{n=1}^{\infty}\frac{3}{n(n+3)}$$ The solution ...
0
votes
1answer
1k views

Volume of the solid whose base is a triangular region with squares as a cross-section

Find the volume of the solid whose base is a triangular region with vertices (0, 0), (2, 0) and (0, 2) if the cross-sections perpendicular to the Y -axis are squares. My problem is that I can't ...
-1
votes
2answers
96 views

Integrating e to higher powers

Say I am to integrate $\int\!2xe^{x^2}\,\mathrm{d}x$ What applies here? I know that when I differentiate, I take the derivative of the inside function times the derivative of the outside function, ...
2
votes
0answers
68 views

how to solve this indefinite integral?

could anyone help me how to solve this indefinite integral? $\int{dx\over \sqrt{\sin^3 x+\sin (x+\alpha)}}$ Thank You.
2
votes
3answers
181 views

$\int_{0}^{\frac{\pi}{2}}\cos{x}\sqrt{1+\tan{x}}dx$

Find this integral $$I=\int_{0}^{\frac{\pi}{2}}\cos{x}\sqrt{1+\tan{x}}dx$$ My try:I find this wolf can't find it. ** then I try: let $$\sqrt{1+\tan{x}}=t\Longrightarrow x=\arctan{(t^2-1)}$$ so ...
2
votes
1answer
51 views

Integrate over a curve (complex)

Given $${\Gamma} = [{z(t) = t\sqrt{\frac2\pi\}}e^{it^2}}]$$ for $$0< t < \sqrt{\pi/2}$$ evaluate $$\int_{\Gamma} ze^{z^2}dz$$ The usual process involves parametrization and then ...
0
votes
1answer
48 views

How to differentiate the equation under the integral sign

i got this question while i was solving the past year papers. I read the posts about the differentiation under the integral sign, but i cannot understand fully and apply them to the equation i got ...
1
vote
1answer
111 views

Evaluating $\int \frac{sin^2(x)}{\sqrt{cos(x)}} \mathrm dx$

I would like to get some advice how to evaluate the integral, $$\int \frac{\sin^2{x}}{\sqrt{\cos{x}}} \mathrm dx$$
3
votes
1answer
55 views

Semicircle contour for integrating $t^2/(t^2+a^2)^3$

Let $a\in\mathbb{R}$. Evaluate $$\int_0^{\infty}\dfrac{t^2}{(t^2+a^2)^3}dt$$ The function is even, so the value of the integral is half of $\int_{-\infty}^{\infty}\dfrac{t^2}{(t^2+a^2)^3}dt$ I'm ...
3
votes
1answer
107 views

Complex integral $1/(z^2+1)$ along circle $|z|=2$

I want to compute the complex integral $$\int_{|z|=2}\frac{1}{z^2+1}dz$$ I write it as a partial fraction $\dfrac12i\int_{|z|=2}\dfrac{1}{z+i}dz-\dfrac12i\int_{|z|=2}\dfrac{1}{z-i}dz.$ Let $f(z)=1$. ...
3
votes
2answers
131 views

Numerical integration for $\int_{0}^{1}f(x)\,dx$

Problem: If $f(x)$ is a polynomial of 5th degree (or less) then show that $$\int_{0}^{1}f(x)\,dx = \frac{1}{18}\left\{5f(\alpha) + 8f\left(\frac{1}{2}\right) + 5f(\beta)\right\}$$ where $\alpha, ...
-2
votes
1answer
86 views

If $\mathrm I_{k} = \int_{0}^{1}\cos(2k{\pi}\,\sin(\,2{\pi}x\,))\,{\rm d}x$ Show $\mathrm I_{k=0}=1,\mathrm I_{k\neq0}=0$ [closed]

Let $ \displaystyle{\mathrm I_{k} = \int_{0}^{1}\cos\left(\vphantom{\Large A}% 2k{\pi}\,\sin\left(\,2{\pi}x\,\right)\right)\,{\rm d}x .\quad}$ Show that 1.$\, k = 0\,,\quad \mathrm I_{k} = 1$. 2.$\, ...