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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0
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2answers
57 views

Euler substitution integrate $\int\sqrt{4x^2+5x+6} dx$

can you help me with integrate this math problem $$\int\sqrt{4x^2+5x+6} dx$$, I tried with 1. Euler substitution, but I stopped on this. $$\int\frac{t^2-6}{5-4t}\frac{-2t^2+5t-12}{5-4t}\frac{10t-4t^...
0
votes
0answers
27 views

Changing the Order of Double Integrals and Evaluating

I have the following problem: I don't know if I'm doing the problem properly because of the "be clever" thing at the end. Each time I do it, I'm getting an answer of 1. Is there something I'm ...
2
votes
0answers
104 views

How to find $\int_0^1\!\! \left(\frac1{\log^2\left(1-x\right)}-\frac1{x^2}+\frac1x-\frac1{12}\right) \frac{\mathrm{d}x}x$

How to show that $$\int_0^1\!\! \left(\frac1{\log^2\left(1-x\right)}-\frac1{x^2}+\frac1x-\frac1{12}\right) \frac{\mathrm{d}x}x=\frac{\ln{(2\pi)}}{12}-\frac{5}{24}+\frac{1}{2\pi^2}\sum_{n=1}^\infty \...
1
vote
1answer
39 views

Jacobian of the Transformation Problem, Multivariable Calculus

I have the following Jacobian problem: I'm having trouble working through it because the double integral in terms of u and v is throwing me off. Could someone walk me through it? Thanks!
3
votes
4answers
105 views

Evaulate $\int_0^1 x\sqrt{\frac{1-x^2}{1+x^2}}dx $

I have been challanged by my teacher to solve this integral, However he gave me no hints, and I have no idea how to start $$\int_0^1 x\sqrt{\frac{1-x^2}{1+x^2}}dx $$ I noticed that putting $x=1-x$ ...
0
votes
0answers
13 views

When Is This Technique For Dealing With Integral Singularities Valid?

I was reading the wonderful paper "Some Series of the Zeta and Related Functions" by V. Adamchik and H.M. Srivastava last night and came across an interesting technique for dealing with singularities ...
4
votes
5answers
85 views

Evaluate $\int e^{2\theta} \sin (3\theta)\ d\theta$ [duplicate]

Evaluate $$\int e^{2\theta} \sin (3\theta)\ d\theta .$$ I am little stuck as to what I can do after this point. Please tell me if my method overall is flawed:
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votes
0answers
25 views

Matrix Integral Problem, what should I do

$A$ and $B$ are square matrices $$\exp(t(A+B))=\exp(tA)+\int_0^t \exp((t-s)A)B\exp(s(A+B))\,\mathrm ds$$ I found it from problem sets. It seems to define a new kind of matrix operation. I can check ...
1
vote
1answer
61 views

Prove $\int\limits_{]0,\infty[}\frac{\ln{x}}{x^2-1} d\lambda_1(x)=\frac{\pi^2}{4}$

I try to prove the following statement: $$\int\limits_{]0,\infty[}\frac{\ln{x}}{x^2-1} d\lambda_1(x)=\frac{\pi^2}{4}$$ There is also a clue: $$ \frac{1}{(1+y)(1+x^2y)}=\frac{1}{x^2-1}\left(\frac{x^2}{...
1
vote
0answers
34 views

Integrating functions

Integrate $f(x)$ from 0 to 1 where $f(x) = \frac{x^3-1}{lnx}$ I received this problem and a variety of others in an advanced mathematics exam. I tried a classical trigonometric substituition approach ...
1
vote
0answers
7 views

Integration that results to give bessels function

I have to integrate the following : $$\int_0^\infty x^{s-1}e^{(-ax^p-bx^{-q})}dx$$ Please help me I am not able to solve this .. However there is this identity available in a book $$\int_0^\infty ...
0
votes
2answers
19 views

Particle's displacement after 4 seconds of motion

Task The velocity, v(t) $ms^{-1}$, of an object travelling along a straight line, at time $t$ seconds is given by: $$v(t)=10e^{-\frac{1}{2}t}sin(\frac{\pi}{2}t)$$ What is the particle's ...
4
votes
3answers
123 views

How to find $\int_0^{2\pi}\log(\alpha+\beta\cos(x))\mathrm{d}x$

Is there a closed-form formula for the following integral $$ \int_0^{2\pi}\log(\alpha+\beta\cos(x))\mathrm{d}x $$ where $\alpha$ and $\beta$ are constants which assure that $\alpha+\beta\cos(x)>0$...
1
vote
1answer
63 views

Integrate $\int\frac{x+1}{(x^2+7x-3)^3}dx$

How can i solve something like that? $$\int\frac{x+1}{(x^2+7x-3)^3}dx$$ How should I start? Should I try rewrite it in partial fractions?
2
votes
1answer
83 views

Understanding how to compute the polygonal image of this Schwarz-Christoffel mapping?

The problem statement reads: This function $\large (−z)^\frac{2}{3} (resp., (1−z)^\frac{2}{3})$ is determined as to be real and positive when $z=x<0$ (resp. when $z=x<1$) and analytic in the ...
0
votes
1answer
32 views

What's wrong with this integral calculation?

I want to calculate the integral $$I = \int_0^{2 \pi} \sin^2 \theta\ \cos^4 \theta\ d \theta$$ by converting it into a complex integral around the unit circle. I use the identities $$\cos \theta = {...
2
votes
0answers
44 views

integration resulting in bessels function [closed]

I have to integrate the following : $$\int_0^\infty x^{s-1}e^{(-ax^p-bx^{-q})}dx$$ Please help me I am not able to solve this .. However there is this identity available in a book $$\int_0^\infty ...
1
vote
1answer
34 views

If $f$ is Riemann integrable and $g$ is continuous, what is a condition on $g$ such that $g \circ f$ has the same discontinuity set as $f$?

I know that if $f$ is Riemann integrable and $g$ is continuous, then the discontinuity set of $g \circ f$ is contained in the discontinuity set of $f$. How would I go about finding a sufficient ...
3
votes
0answers
89 views

Convergence for a improper integral $\int^b_a fg$

Let $f$ be continuous on [a,b) such that $\int^b_a f$ converges. If $g'$ is locally integrable and has a constant sign on [a,b), prove that $\int^b_a fg$ converges. Edit: We can assume that the limit ...
0
votes
1answer
36 views

integral constant next to theta

I am a bit stuck on this practice question for an upcoming integration exam (calculus 1). Here is the integral: $$\int_ {0}^{\frac{\pi}{2}}{2\cos(3\theta)d\theta}$$ What confuses me is the 3 next ...
0
votes
2answers
51 views

Calculating the derivative with limited info.

$$G(x) := \int_x^{x^2} f(t) \ dt$$ Calculate G'(x). I've made some progress by integrating by parts with f(t) = 1(f(t)) but I'm stuck now and don't know where to go.
0
votes
2answers
54 views

Definite Integral involving reciprocals of logs

Integrate $$\int_2^{4e} \frac{1}{x \ln(x+1)}\,dx $$ I have tried partial fractions, u substitution and parts but i cant get the final answer out. my main problem is dealing with the $x$ and $x+1$ ...
2
votes
1answer
63 views

Evaluate the following integral $\int_{-1/2}^{1/2}\big(\frac{\sin(n\pi f)}{\sin(\pi f)}\big)^4 df$

There are similar questions out there, but I was hoping someone could show how to would evaluate the following integral $$\int_{-1/2}^{1/2}\bigg(\frac{\sin(n\pi f)}{\sin(\pi f)}\bigg)^4 df$$ I've ...
0
votes
1answer
32 views

Length of the curve $ \;\;x=3\cos\!\left(6t\right), \;\;y=18t+3\sin\!\left(6t\right), \, \;\; \;\; 0 \le t \le \frac{\pi }{6}\;\;$

The length $\;L\;$ of the curve C given by $\displaystyle \;\;x=3\cos\!\left(6t\right), \;\;y=18t+3\sin\!\left(6t\right), \, \;\; \displaystyle \;\; 0 \le t \le \frac{\pi }{6}\;\;$ is found by ...
3
votes
2answers
66 views

Integral of $\frac{1}{x^2+4}$ Different approach

underneath is a brief method of partial fractions integration on the problem given in the title Using a standard trigonometric result it is known that: $$ \int \frac{1}{x^2+4}dx=\frac{1}{2}\tan^{-1}(\...
1
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0answers
37 views

About the integral $\oint \frac{z+\bar{z}}{\left | z \right |}dz$

Suppose we wanted to integrate this around a semicircular path above above the for $\Im \left ( z \right )\geq 0$ and $\left | z \right |\leq e$. The integrand has an essential singularity at $z=0$, ...
3
votes
1answer
43 views

Probability Density Function Equation, Multivariable Calculus

I have the following problem: The formula for the normal distribution has a π in it. In this simplified version of the normal probability density function, solve for C. The correct answer has π in it....
1
vote
1answer
67 views

Is it always true that the Lebesgue integral of a continuous function is equal to the Riemann integral (even if they are both unbounded)?

Let's assume that $f\colon\mathbb R\to\mathbb R$ is continuous and hence Lebesgue measurable. Then, the Lebesgue integral $\int_{(0,\infty)}f(x)\,d\lambda(x)$ makes sense (but of course can be equal ...
0
votes
3answers
88 views

Calculate $\int_{-\infty}^{\infty} e^{-x^2}\cos (ax) dx$ using Taylor series cosine

Let $a > 0$. Im trying to show that $\int_{-\infty}^{\infty} e^{-x^2} \cos (ax) dx = \sqrt{\pi}e^{-\frac{1}{4}a^2}$. I'm taking a course on measure theory, and I want to prove this using the ...
2
votes
2answers
42 views

Prove that $\int_{0}^{1/2} x^{-\alpha}|\log(x)|^{-2\alpha} d \, x$ diverges for $\alpha>1$

As a part of a larger proof I need to show that $\int_{0}^{1/2} x^{-\alpha}|\log(x)|^{-2\alpha} d \, x$ diverges for $\alpha>1$ (which I know to be true from numerical simulation with Maple). ...
0
votes
2answers
54 views

Hint for integrating exp(x-x^2)

The function $e^{x-x^2}$ is zero if $x \to \infty$ or $x \to -\infty$ it looks like a normal-distribution-curve with the max. value at $x=0.5$. Has somebody a hint for integrating it from $-\infty$ ...
1
vote
4answers
66 views

What is the integral of $\frac{x-1}{(x+3)(x^2+1)}$?

I've worked with partial fractions to get the integral in the form $$\int\frac{A}{x+3} + \frac{Bx + C}{x^2+1}\,dx$$ Is there a quicker way?
1
vote
2answers
878 views

Find the area of the region that lies inside the first curve and outside the second curve. $r = 10 \cos\theta,\ r = 5$

I am not sure of my answer. In the figure, $r=10 \cos\theta$ is a circle that doesn't look like a circle. The area of $r=5$ is $\pi r^2 = 25 \pi$. You remove the area from $-\pi/3$ to $\pi/3$ of $...
1
vote
1answer
59 views

How the tackle this limited integral?

I started with use of a new variable for the things under square root. I would like to calculate the integral $$\iint_\Omega\sqrt{\sqrt{x}+\sqrt{y}}~\mathrm{d}x~\mathrm{d}y$$ over the respected area ...
1
vote
1answer
64 views

Complex Integrate $\int_{-\infty}^{\infty}e^{-|\lambda t|}e^{itx}dt$

I'm working through Big Rudin's (Real and Complex Analysis) Fourier Transform chapter, and the following complex integral is part of a discussion on the Inverse Transform that Rudin mentions briefly ...
0
votes
1answer
61 views

Three point Gaussian Quadrature formula derivation

For 3-point Gaussian quadrature, I'm not sure how the $5/9$ and $8/9$ coefficients are found. I am able to derive $x0, x1, x2$ in $g(x0), g(x1), g(x2)$ but I'm not sure how to get the rest.
1
vote
1answer
34 views

How to find the limits of integration of this double integral using Iverson Brackets?

I'm looking at the approach laid out over here, and have some steps about how to get the limits of integration using Iverson brackets. I want to find $$\int \int f(x,y)\ dy\ dx$$ where $$x \geq ...
1
vote
3answers
67 views

How to integrate $\int_{0}^{\infty}{\cos(tx)x^{\alpha-1}e^{-x/\lambda}}dx$ and $\int_{0}^{\infty}{\sin(tx)x^{\alpha-1}e^{-x/\lambda}}dx$

The original problem is $\int_{0}^{\infty}{e^{itx}x^{\alpha-1}e^{-x/\lambda}}dx$ My work: $\int_{0}^{\infty}{\cos(tx)x^{\alpha-1}e^{-x/\lambda}}dx=-\lambda\int_{0}^{\infty}{\cos(tx)x^{\alpha-1}de^{-...
0
votes
2answers
46 views

Definite integral $1/(t(1-t))^{3/2} \exp(-a/t-b/(1-t))$ [closed]

I'm trying to find the result of the following definite integral $$ \int_0^1 \frac{d t}{(t(1-t))^{3/2}}\, \exp\left(-\frac{a}{t}-\frac{b}{1-t}\right)$$ for positive $a$, $b$. Any hints or possible ...
3
votes
1answer
152 views

How To Compute $\int_{-\frac12}^3 \frac{\frac1{x+1}\ln (x+2)}{\ln(x+3)+\ln(x+2)-\ln(x+1)}dx$?

Evaluate the following integral $$\int_{-\frac12}^3 \frac{\frac1{x+1}\ln (x+2)}{\ln(x+3)+\ln(x+2)-\ln(x+1)}dx$$ The answer in a closed form seems to be $\frac32 \ln2$.
1
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0answers
22 views

Convergence of Integral of a Function that does not converge

Recently, I thought about the following: Can you say if an integral will or won't converge just by considering the function itself? For example, can an integral converge if the function you're ...
1
vote
2answers
35 views

Showing convergence of integral

I need to proof that the following integral exists: $$\int_1^\infty t^{a+\sigma-1} e^{-t} \, dt \text{ for every }0<\sigma<a$$ However, I can´t find a proper upper bound for that integral. ...
1
vote
1answer
18 views

How do I calculate the Fourier-transform of $f(t) = \max(t-1,0) (t \in \mathbb{R})$?

I get $$\hat f(w) = \int_{-\infty}^{+\infty}\max(t-1,0)e^{-i\omega t}dt$$ $$= -\int_{-\infty}^{1}(t-1)e^{-i\omega t}dt$$ $$ = \lim_{p\to\infty}\left(\int_{-p}^1e^{-i\omega t} - \int_{-p}^1 t\cdot e^{-...
1
vote
0answers
45 views

Integrate exponential with modulus

I need to prove the following integral relation: $$\lim_{T \to (1-i \epsilon) \infty} \mathrm{\int_{-T}^T dt_1 dt_2 e^{-ia |t_1|} \cdot e^{-ib |t_2|} \cdot e^{-ic |t_1-t_2|}} = \frac{-2}{(a+b)(b+c)} + ...
3
votes
3answers
232 views

Evaluating A Path Integral In Polar Coordinates

Show that the path integral of $f(x,y)$ along a path given in polar coordinates by $r=r(\theta)$ where $\theta_1 ≤ \theta ≤ \theta_2$, is $$\int_{\theta_1}^{\theta_2} f(r \cos \theta ,r \sin \theta) \...
0
votes
2answers
192 views

Volume bounded by elliptic paraboloids

Find the volume bounded by the elliptic paraboloids given by $z=x^2 + 9 y^2$ and $z= 18- x^2 - 9 y^2$. First I found the intersection region, then I got $x^2+ 9 y^2 =1$. I think this will be area of ...
1
vote
0answers
26 views

Prove convergence in $L^1$

Let $(X, \mathscr{A}, \mu)$ be a finite measure space. Let $f_n \in L^1$. Assume $f_n \rightarrow f$ a.e. and there exist $p > 1$ and $c > 0$ such that $$||f_n||_p < c$$ for all $n$. I want ...
0
votes
1answer
47 views

Testing the convergence of an improper integral.

Find all the real values of $p$ and $q$ so that $$ \int_{0}^{1} x^{p}\biggl(\ln\frac{1}{x}\biggl)^q dx$$ converges. I tried using comparison test but couldn't solve it. Please help me.
1
vote
1answer
62 views

Mutual information for a continuous uniform distribution

I'm trying to compute using matlab the mutual information for an $ \infty $-PAM input (the limit of a very dense PAM constellation) for a range of snr and I got stuck. I'm working with a real-valued ...
3
votes
1answer
48 views

How do I express $\int_0^{\frac{\pi}{2}}\sin^{2m-1}\left(t\right)\cos^{2n-1}(t)dt$ using the Gamma-function?

We define the Gamma function as: $$\Gamma(p)=\int_0^{+\infty}e^{-x}x^{p-1}dx$$ I was advised to rewrite the integral as $\sin(t)^{2m-1}\cos(t)^{2n-2} d \sin(t)$, and substitute $ t = \sin(t)$ which ...