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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3
votes
1answer
92 views

How can I do this? $\int\frac{dx}{x^4+1}$ [duplicate]

I tried to integrate this: $\displaystyle\int \dfrac{dx}{x^4+1}$ I tried to do it with the partial fractions method (after factoring the denominator), but the process is really large, and I got a lot ...
2
votes
2answers
40 views

Find volume of cask

I was given the following question: A wine cask has a radius at the top of $30 cm$ and a radius at the middle of $40 cm$. The height of the cask is $1m$. What is the volume of the cask in litres, ...
1
vote
1answer
53 views

System of ODE - Solution

I have a system of ODE to solve $$ A_{5 \times 5}\ddot{q}(t)_{5 \times 1}+ B_{5 \times 5}\dot{q}(t)_{5 \times 1}+ C_{5 \times 1} =0\tag 1$$ Given Data $A,B,C$ are constants.We know what is ...
3
votes
2answers
42 views

Find arc length of curve on the given interval

I was asked to find the arc length of the curve of the following curve: $24xy = x^4 + 48$ from $x = 2$ to $x = 4$ This has turned out to be a very difficult problem, I get stuck using the arc length ...
0
votes
0answers
12 views

Definite integral - Finding an equivalent form

I have the following definite integral $ \int_{0}^{L} {\psi(t) }_{1 \times 5}{A(s)}_{5 \times 5}(\psi(t) _{1 \times 5})^{T} {B(s)}_{5 \times 5} ds \tag 1 $ Given data All dimensions are ...
1
vote
1answer
56 views

Integrating $(2x+1)^{-2}$

How do I integrate the following expression: $$(2x+1)^{-2}$$ I should end up with something like this: $$\frac{-1}{2(2x+1)}$$
1
vote
1answer
36 views

If a real polynomial of degree $n\gt 1$ has a root of modulus exceeding all others, is that one a real root?

Suppose $a_nx^n+\ldots+a_1x+a_0=0\; (a_n\in \mathbb{R})$ has $n$ distinct roots $r_1,r_2,\ldots, r_n$ (no multiple roots), and if $\exists r_k$ s.t. $\forall r_i\in\{r_1,r_2\cdots r_n\}-\{r_k\}$, ...
6
votes
0answers
151 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
4
votes
5answers
190 views

Integral of $1/[(1+x^2)\sqrt{1+x^2}]$

I try to get back on track with the integration. I would like to solve $$ \int_0^1 \frac{dx}{(1+x^2)\sqrt{1+x^2}}.$$ There are my way to try to solve it (that I don't find the right solution) and an ...
3
votes
1answer
44 views

Solving linear non-homogeneous integral equation

Is it possible to solve equations of the kind: $$x(t) = \int \limits_t^T c_1 x(s) ds +c_2 $$ with $c_1$ and $c_2$ being some constants and if I know $x(0)$? Or do I need more assumptions on $x$?
1
vote
1answer
108 views

How can I express $\int \frac{1}{f'(x)}$ in terms of $f(x)$

More specifically, I would like to know if there is a way I can express $$\int \frac{x g'(x)}{f'(x)} dx $$ In terms of $f(x)$ and $g(x)$. Both $f(x)$ and $g(x)$ are non-negative and known to be ...
2
votes
2answers
79 views

Integration of $1/\sin^3 x$

I need a explanation of this problem: $$ \int \frac{1}{\sin^3 x}\,dx $$ Change the variable $$ t = \tan (x/2) $$ With use of $\tan$, $\cos$, $\sin$ and $\cot$, only. So how do I ...
1
vote
1answer
51 views

$dxdy=-dydx$ using Jacobian determinant. Why?

How do you reslove the contradiction due to the fact that $dxdy = dydx$ as per definiton of hyperreals ? Is this abuse of notation and by $dxdy$ its is actually meant $dx \wedge dy$ in both ...
3
votes
1answer
48 views

Finding all the possible values of an Integral in the Complex Plane

I am studying Complex Analysis by Lars V Ahlfors. I am unable to solve one of his exercises. It is: Find all possible values of $$\int \frac{dz}{\sqrt{1-z^2}}$$ over a closed curve. I do not have ...
1
vote
2answers
54 views

Prove Schwarz inequality in $R^2$

Can someone please show me how you would prove the following in $R^2$ $\int f(x)* g(x) dx \leqslant \int f(x)^2 dx * \int g(x)^2 dx $ starting from $\int [\lambda*f(x) - g(x)]^2 dx \geqslant ...
1
vote
6answers
73 views

Initial value $\left ( \frac{dy}{dt} \right )+3y=11$, $y(0)=1$

I have never done an initial value problem, and would like some help on how to start this please.
0
votes
4answers
97 views

Definite integral $\int_{-64}^{1}\frac{dx}{x^{1/3}}$

I am having some trouble with a problem very similar to this in my study guide, how can I start, the $-64$ is really intimidating to me.
1
vote
2answers
50 views

Relating $\sin(x)^{2a+1}$ on interval $(0,\pi/2)$ to a factorial form

The integral: $$\int_0^{\pi/2} \sin(x)^{2a+1}dx $$ has a closed form solution in terms of factorials: $$[(2^a)(a!)]^2 [(2a+1)!]^{-1}$$ How does this come about?
2
votes
0answers
61 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
1
vote
1answer
45 views

Matrix - Commutative property

I have a rotation matrix represented as $R(t)=e^{B(t)},\tag 1$ where $B(t)$ is a skew symmetric matrix (since any rotation matrix can be expressed as a matrix exponent of a skew symmetric matrix), ...
5
votes
0answers
44 views

How to compute product integrals?

From the wikipedia article about product integrals I can see that if our function is scalar, then to compute type I product integral we can just take exponential of a usual integral: $$\prod_a^b ...
2
votes
0answers
32 views

Integration indefinite integral of multiple functions

I need help integrating $$\frac{x}{1-e^{-(x/a)^2}}\times e^{\frac 12 \left(\frac{x-u}{s}\right)^2}$$ wrt $x$ where $a$ and $u$ are constants
-1
votes
0answers
25 views

The probability of a 2n-polynomial with random coefficients has real root? [closed]

How to computer the probability ,Beside , what is the probability of the real root greater than 1?
0
votes
0answers
30 views

Derivation using Ito calculus?

I am reading the paper "Coupling Wiener processes by using copulas" by P. Jaworski and I've come across a statement I cannot reproduce. Let $L^{-}$ and $L^{+}$ be differential operators acting on ...
0
votes
0answers
32 views

Equation derived by integration by parts?

I am reading the paper "Coupling of Wiener processes by using copulas" by P.Jaworski and M.Krzywda and as I am reading the proofs, one derivation I can't quite understand. Let $L^{-}$ and $L^{+}$ be ...
1
vote
1answer
26 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi ...
1
vote
3answers
139 views

Differential equation $\sin \theta \frac{dr}{d \theta}+r\cos \theta =\tan \theta,0<\theta<\pi/2$ [closed]

This problem has been stumping me for over an hour how can I set it up, I think I have done it wrong over and over. Solving for $r$.
2
votes
4answers
50 views

How to solve $(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$

Solve $$(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$$ I have a very similar problem like this on my homework, and I have no clue how to set it up or even start. How could I set ...
0
votes
2answers
43 views

Calculus long division $\int\frac{y^4+3y^2-1}{y^3+3y}\ dy$

I have a problem like this in my homework and want to see how to go by doing this problem. I understand the long division, but cannot get the partial fraction part. $$\int\frac{y^4+3y^2-1}{y^3+3y}\ ...
1
vote
3answers
78 views

Why does solving $\int \frac{v}{9.8-0.0025v^2}\mathrm{d}v=\int1{d}x$ for $v^2$ in terms of $x$ produce 2 completely different answers?

In this question $g=9.8$ (acceleration of free fall). You are also given that when $x=0$ $v=0$. My answer is $v^2=400g(1-e^\frac{x}{200})$. I obtained it by integrating both sides so that ...
0
votes
1answer
46 views

Need an example/counterexample of continuous and increasing function.

If $\mu$ is a finite measure on the measurable space $\big( X, \mathscr{F} \big)$, $f : X\to [ 0, +\infty)$ is measurable. Then $\textbf{does it exist a continuous function $g : [ 0, +\infty)\to [ ...
0
votes
3answers
159 views

Integration of $e^{\cos x}\cos x$ [on hold]

Could you please help me to solve this integration problem? I could not find an exact symbolic expression for it. $$\int {{e^{\cos x}}} \cos xdx$$
2
votes
2answers
84 views

Any idea how to linearize this equation? $X^2-Y^2=aZ+bZ^2$

The intention is to linearize this equation $X^2-Y^2=aZ+bZ^2$ into something which looks like $Z=mX+nY+c$ so that a graph of $Z$ against $X$ or $Y$ can be plotted. X,Y,Z are variables while a,b,c are ...
4
votes
2answers
124 views

Definite integral $\int_{0}^{\infty}e^{-u}\frac{1}{\left(\sqrt{1+(h+u)^{2}}\right)^{5}}du$

Hi guys I have the following definite integral to solve: $$\int_{0}^{\infty}e^{-u}\frac{1}{\left(\sqrt{1+(h+u)^{2}}\right)^{5}}du$$ is it possible to obtain an analytic expression? And if not why? ...
1
vote
0answers
35 views

Verifying an antiderivative found in any integral table

If $a > 0$, and $0 < b < c$. \begin{equation*} \int \frac{1}{b + c\sin(ax)} \, {\mathit dx} = \frac{-1}{a\sqrt{c^{2} - b^{2}}} \, \ln\left\vert\frac{c + b\sin(ax) + \sqrt{c^{2} - ...
1
vote
1answer
96 views

Can you find integral of this function.

Question: Consider $$F(x) = \frac{1}{\sin(x-a) \ \sin(x-b) \ \sin(x-c)}$$ Then, how to compute $\int F(x) \, \mathrm{d}x$? Edit: I have tried what I know about integral solving methods. I ...
1
vote
1answer
61 views

Integration question.

The question is as follows For any real number $x$, let $\lfloor{x}\rfloor$ denote the greatest integer less than or equal to $x$. Let $f$ be a real valued function defined on the interval ...
0
votes
3answers
63 views

Area of the region: $\;x ≥ 0; \;−x\sqrt3 ≤ y ≤ x\sqrt3;\,\;(x−1)^2 + y^2 ≤ 1$.

Can anyone please explain how to set up the needed integral? I need to calculate the area of the region given by: $x ≥ 0,$ $-x\sqrt3 ≤ y ≤ x\sqrt3,$ $(x−1)^2 + y^2 ≤ 1$.
2
votes
3answers
196 views

Why are variables in integration by substitution so counter intuitive?

Integration by substitution is defined as something like $\displaystyle\int_a^b f(\phi(t))\phi'(t)dt = \int_{\phi(a)}^{\phi(b)} f(x)dx$ But for my taste, the variables $x$ and $t$ are exactly ...
2
votes
0answers
52 views

Can this modified Gaussian integral be calculated analytically?

In my research, I encounter this modified Gaussian integral $$\int_{-\infty}^{\infty}dx\,\frac{x+\sqrt{x^2-bx}}{2\sqrt{x^2-bx}}\exp\left[-a^2(x-x_0)^2+i\left(cx-d\sqrt{x^2-bx}\right)\right],$$ where ...
0
votes
1answer
42 views

Integral of two function multiplied [closed]

Please let me know what / how does the result of below integral results would differ: $$(f \star g)(\tau) = \int_{-\infty}^\infty f(t)\star g(t+\tau)\,dt,$$ $$(f + g)(\tau) = \int_{-\infty}^\infty ...
2
votes
0answers
43 views

Homotopy, Stokes Theorem and Orientation

I have a problem in which the theory and the computation disagree about a minus sign. My question requires a little setting up. I have a complex valued 2-form $$ \omega = \alpha(\xi_1,\xi_2)\, ...
-4
votes
0answers
38 views

determine the answer of this integration. [closed]

I was wondering what the answer is to the following integration problem? $$\int\frac{dx}{\sqrt{1+\frac{b}{x^2-a^2}}}$$ Here, $a$ and $b$ are constants.
2
votes
2answers
210 views

An Improper Integral

I need help with this integral: $\Large {\int_0^\infty \frac{dx}{x\sqrt{1+x}}} $ What I did: Substitute $\sqrt {1+x} = t$. Then the integral turns into $ \int_1^\infty 2dt/(t^2-1) $. Now I replaced ...
1
vote
3answers
57 views

How to calculate the integral of $\sum_{n=1}^\infty (1/r)^{n+1} r^2$?

How to calculate this integral? $$\int\limits_0^1 {\sum\limits_{n = 1}^\infty {\left( {\frac{1}{r}} \right)} } ^{n + 1} r^2 dx$$ Here $r$ is a real number
2
votes
0answers
33 views

Projection measures and integrability

Let $(M, \mathcal{A}, \mu)$ a probability space, $Y$ compact metric space. Consider $\mathcal{M}(\mu)$ be the space of probability measures $\eta$ on $M\times Y$ such that $\pi_{*}\eta=\mu $ where ...
1
vote
1answer
16 views

Reference on Riesz representation theorem for $L^p(0,T,X)$ spaces.

Brezis Functional Analysis book proves the following Riesz representation theorems for usual $L^p(\Omega)$ spaces: In what book can we find an analogous of these theorems for $L^p(0,T,X)$ spaces? ...
1
vote
1answer
32 views

Using Stokes' Theorem to evaluate an integral around a triangular path

Problem: Use Stokes’ theorem to evaluate the integral $I = \int\limits_C \textbf{F} \centerdot \textbf{ds}$ when $\textbf{F}$ is the vector field $\textbf{F} = 3zx\textbf{i} + 3xy\textbf{j} + ...
0
votes
0answers
41 views

Integrating by parts with exponential and power-law functions

I have a question about integrating by parts for $$\int_{L}^{U}\left[x^{a} \cdot e^{-bx}\right]\,dx$$ for positive reals $L,U$ with $L<U$ ($L, U \in [0, +\infty) $). I'm interested in cases with ...
1
vote
2answers
92 views

How can I calculate $\int \frac{\sin kx}{a+bx}\;dx$?

How can I calculate $$\int \frac{\sin kx}{a+bx}\;dx \quad ?$$ I know its final solution but I don't know its solution step by step. Thanks.