Tagged Questions
All aspects of integration, including the definition of the integral and computing indefinite integrals (antiderivatives).
70
votes
6answers
4k views
How can you prove that a function has no closed form integral?
I've come across statements in the past along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations:
addition/subtraction
...
16
votes
6answers
889 views
Closed form of integral.
I've been looking at
$$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$
It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example:
...
23
votes
6answers
3k views
Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral?
In this wikipedia article for improper integral,
$$
\int_0^{\infty}\frac{\sin x}{x}dx
$$
is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) ...
43
votes
6answers
4k views
Ways to evaluate $\int \sec \theta \, d \theta$
The standard approach for showing $\int \sec \theta \, d \theta = \ln |\sec \theta + \tan \theta| + C$ is to multiply by $\frac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a ...
11
votes
1answer
723 views
If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$?
Let $f$ be a measurable function on a measure space $X$ and suppose that $fg \in L^1$ for all $g\in L^q$. Must $f$ be in $L^p$, for $p$ the conjugate of $q$? If we assume that $\|fg\|_1 \leq C\|g\|_q$ ...
42
votes
4answers
1k views
Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even
I have a question:
Suppose $f$ is continuous and even on $[-a,a]$, $a>0$ then prove that
$$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$
How can I ...
2
votes
2answers
280 views
Complex Fourier series
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
7
votes
4answers
585 views
How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$?
I have to compute this integral $$\int_{0}^{\infty} \frac{dt}{1+t^4}$$ to solve a problem in a homework. I have tried in many ways, but I'm stuck. A search in the web reveals me that it can be do it ...
4
votes
3answers
219 views
Indefinite integral of secant cubed
I need to calculate the following indefinite integral:
$$I=\int \frac{1}{\cos^3(x)}dx$$
I know what the result is (from Mathematica):
$$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$
but I don't ...
3
votes
2answers
997 views
How to integrate $\int e^{-t^{2}} \space dt $ using introductory calculus methods
Earlier today I stumbled across this when I was doing some practice questions for a physics course: $$\int e^{-t^2} \space dt $$
To expand, the limits of integration were something like $1$ and $4$ ...
34
votes
5answers
1k views
Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$
I've got troubles in computing the below integral:
$$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$$
I hope it can be expressed in elementary functions. I've tried simple substitution as $u=\sin(x)$ ...
16
votes
5answers
898 views
Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$
I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general:
$$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$
...
5
votes
4answers
358 views
How does partial fraction decomposition avoid division by zero?
This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example:
$$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$
Multiplying ...
6
votes
4answers
1k views
$f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$
This is problem 6.2 from the 3rd edition of Principles of Mathematical Analysis.
Problem 6.2: Suppose $f\geq 0$, f is continuous on $[a, b]$, and $\int_{a}^{b}f(x)dx = 0$. Prove that $f(x)=0$ for ...
5
votes
4answers
414 views
Finding $\int e^{2x} \sin{4x} \, dx$
Finding $$\int e^{2x} \sin 4x \, dx$$
I think I should be doing integration by parts...
If I let $u=e^{2x} \Rightarrow du = 2e^{2x}$,
$dv = \sin{4x} \Rightarrow v = -\frac{1}{4} \cos{4x}$
$\int{ ...
