5
votes
1answer
47 views

Maximum value problem

A function $\hspace{0.1cm}$$f:[0,1]\to[-1,1]$$\hspace{0.1cm}$ satisfying$\hspace{0.1cm}$ $|f(x)|\leq x$$\hspace{0.1cm}$ $\forall x\in[0,1]$. Then find the maximum value of: ...
5
votes
5answers
172 views

Evaluating the following integral: $ \int \frac{x^2}{\sqrt{x^2 - 1}} \text{ d}x$

For this indefinite integral, I decided to use the substitution $x = \cosh u$ and I've ended up with a $| \sinh u |$ term in the denominator which I'm unsure about dealing with: $$\int ...
0
votes
1answer
50 views

Integral of absolute value = absolute value of the integral

Let $(a,b) \in \mathbb{R}^2$ and $f \in C^0([a, b] , \mathbb{C})$ Find the condition on $f$ so that $$|\int_a^b f|=\int_a^b|f|$$ My try : The function $f: t \mapsto r(t)\exp(i\theta)$ where $r$ is a ...
0
votes
1answer
28 views

Integral of The Absolute Value of x

$$ \int_{1}^4|x|dx $$ I know how to take the integral of a more complex function (like f(x)= |x+2|) but I don't understand what to do if it's just the absolute value of x. If the lowest number is 1 ...
0
votes
2answers
45 views

How to integrate absolute function

I have this absolute e-function, but I don't know how to calculate the integration $$ \int_{-2}^{2} e^{\frac{1}{2}j\omega |x|}dx $$ Any idea?
0
votes
1answer
24 views

Calculate the area that the following graphs form

I have been trying and trying to solve the following problem (I even used wolframalpha as an extra help, but no success, and I have like 100 calculations in my notebook): The Task: Calculate the ...
0
votes
1answer
57 views

Change of variables - integrals

\begin{equation} \text{Let $\hspace{3mm}$ }f(t) = 2\int_{b}^{\infty} \sqrt{\frac{1}{2\pi t}}e^{-x^2/2t}dx. \end{equation} I found that this integral can be written with change of variables can be ...
1
vote
1answer
39 views

Proving a claim $|c_n e^{in\theta}| = |c_n|$

I'm studying about Fourier series from a book called "Fourier series and its applications" by Folland and on page 40, the author makes the claim that: $$|c_n e^{in\theta}| = |c_n|,$$ where $n$ is an ...
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
0answers
108 views

Integration of the absolute value of an unknown function

I'm doing a vector arclength problem, and have gotten to the part where I have $\int | r'(t) | dt. $ Both $r(t)$ and $r'(t)$ are unknown functions, though I do know that $0 \leq r(t)$ for $a ≤ t ≤ ...
0
votes
1answer
73 views

Matrix integral of absolute exponential item

If $A=(a_{ij})$ is an $n\times n$ symmetric positive matrix, is it possible to calculate the following matrix integral? $$\int_{0}^{\infty}\left | e^{-A(t+1))}-e^{-At)} \right |\mathrm dt,$$ where ...
5
votes
2answers
187 views

Absolute values in $\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$

in my math class we were given a list of indefinite integrals, and one of them was: $$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$$ My working: $$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}=\int ...
0
votes
2answers
90 views

Calculating the integral $ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$

How do we calculate the following integral: $$ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$$
3
votes
3answers
317 views

How does one calculate the integral of the sum of two absolute values?

I know how to find the integral of just one absolute value, but this problem presents the integral of the sum of two absolute values. Help! I want to evaluate: $$ \int_a^b{(|x-1| + |x+1|) dx} $$
1
vote
1answer
101 views

Integration Involving the Absolute Function

How do I integrate the double integral of the form $|x^2-y|$ with the boundaries $-1\leq x\leq 1$ and $-1\leq y\leq 1$?
1
vote
1answer
439 views

double integral of an absolute function

I'm just a little unsure of how to tackle this one. I understand that typically you would separate the integral into two for where x is positive or negative, I'm just unsure of how to separate it for ...
3
votes
1answer
104 views

Integral of absolute e-function

I have to integrate the following function: $$\int e^{-|x|}$$ I tried this and I don't think, that this is right. So can you tell me, where my fault is? $$\int e^{-|x|} = ...
4
votes
3answers
748 views

Proving two integral inequalities

Can anyone help me to prove that these integral inequalities hold? Here $x$ is a real value: $$ \left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx $$ Here $z$ is a complex value: $$ \left| ...
3
votes
6answers
333 views

Evaluating $\int |x|^3 \; dx $

$$\int |x|^3 \; dx $$ In my module it is suggest to use integration by parts, $$ \text{ Set }I = \int (|x|^3 \cdot 1) \; dx = |x|^3 \cdot x - \int \color{red}{\frac {x^3}{|x|^3}3x^2}\cdot x \; dx$$ ...
1
vote
2answers
328 views

Smoothing of absolute value and sign functions for numerical integration

I'm doing Numerical integration of ODEs. for a special system that has an always positive coordinate s and a conjugated momentum ...
4
votes
1answer
25k views

Integral of an absolute value function

How do I find the definite integral of an absolute value function? For instance: $f(x) = |-2x^3 + 24x|$ from $x=1$ to $x=4$
4
votes
3answers
486 views

Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?

Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$ Proof (attempt): We know $-|f|\le f \le|f|$, so $\int-|f| \le \int f \le \int|f|$.* Since, if $-b<a<b$, we say ...
4
votes
1answer
683 views

integral from 0 to $2\pi$ of $|\cos x|\operatorname{d}x$ not integrating as I'd expect

I drew a rough sketch of $|\cos x|$ and would guess the correct answer to this integral is $4$ because I know the area under the curve of $\cos x$ from $0$ to $\pi/2$ is $1$, and there are $4$ such ...