-1
votes
3answers
68 views

How to evaluate an integral of the form $\int \frac{dx}{-ax^2 + b}$?

I need to evaluate $\int \frac{dx}{-ax^2 + b}$ while both $a$ and $b$ are positive. I was blocked while I was trying $ x=\tan\theta $ which turned $ dx=\sec^2\theta\, d\theta $ This method didn't ...
4
votes
4answers
247 views

Limit of a rational function

Calculate the limit $$ \lim_{x \to 0} \frac{3x^{2} - \frac{x^{4}}{6}}{(4x^{2} - 8x^{3} + \frac{64x^{4}}{3} )}$$ I divided by the highest degree of x, which is $x^{4}$, further it gave $$ ...
0
votes
2answers
48 views

Integral of $\arcsin$ of a rational function, using integration by parts

I'm a class 12 student and this a question from my textbook: $$I=\int{\arcsin{2x\over 1+x^2}}\mathrm{d}x$$ I did it using integration by parts like this: $$I=\arcsin{\left(2x\over ...
1
vote
3answers
52 views

A simple-looking rational limit

Please help me compute: $$ \lim_{z\to 0}\frac{\sqrt{2(z-\log(1+z))}}{z} $$ I know the answer is 1 because I plugged it into Mathematica. Attempts with L'Hopital's Rule didn't work. This a step in an ...
0
votes
2answers
52 views

Find all values $c$ such that $(x+1)/(x^2+2cx+4)$ has domain R

Word for word: Find all values of $c$ such that $f(x)=\frac{x+1}{x^2+2cx+4}$ has a domain R I really don't know where exactly to start. I'm not sure what it means by "a domain R"
2
votes
2answers
50 views

Integration of the rational function $ 4/( 1+4t^2) $

So it's August so my memory of math is a little rough right now. I was wondering if someone could help me with integration with a fraction involved? For example: $$\int_0^{1/2} \frac 4{1+4t^2} \, ...
1
vote
2answers
34 views

Derivative of rational function help.

consider $$f(x)=\frac{1}{2x-4}$$ The derivative should be $\displaystyle -\frac{1}{2(2x-4)^2}$ However I get $\displaystyle -\frac{2}{(2x-4)^2}$ my workflow: $$\begin{array}{} f'(x)&= ...
5
votes
1answer
113 views

There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
2
votes
1answer
156 views

How to evaluate $\int_0^1\frac{1+x^4}{1+x^6}\,dx$

$$\int_0^1\frac{1+x^4}{1+x^6}\,dx$$ Can anyone help me solve the question? I am struggling with this.
11
votes
2answers
269 views

$\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$

Prove that: $\lfloor n^{1/2}\rfloor+\cdots+\lfloor n^{1/n}\rfloor=\lfloor \log_2n\rfloor +\cdots+\lfloor \log_nn \rfloor$, for $n > 1,\, n\in \mathbb{N}$ For example. For $n=2$, we have $\lfloor ...
0
votes
2answers
60 views

Build the function by its values. Only combination of +, -, *, abs() allowed for this function.

I've decided to open a new, more common question about the simplest function f(1)=-1; f(2)=0; f(3)=1; f(4)=0.. So, here is the question. Let's say we have some function $y=f(x)$ we'd like to find by ...
1
vote
1answer
132 views

Partial Fraction Decomposition of unknown power

I have an equation of the form: $$\frac{1}{(1+as)(1+0.5bs)^m}$$ where $m$ is unknown, and its range is $(1,2,3,...)$ How I can do the partial fraction? I am now reading a paper that derived an ...
3
votes
2answers
64 views

How to find an approximate values of rational function $f(x)$ for large $x$, neglecting $\frac{1}{x^4}$ and successive terms?

This is the function that I want to find an approximate value for it neglecting $\displaystyle \frac{1}{x^4}$ and successive terms $$ f(x)=\frac{25x}{(x-2)^2(x^2+1)}. $$
0
votes
2answers
67 views

Why does only one of these limits exist?

I'm reading Adams' Calculus - A Complete Course and got stuck on something I'm guessing is quite easy. Anyway, I'm wondering why it is that the limit $$ \lim_{x\to 2}\frac{x-3}{(x-2)(x+2)} $$ does ...
0
votes
3answers
593 views

Integral of a product of functions divided by the integral of one of the two functions

Ratio between two integrals: $$\frac {\int f(x)g(x)} { \int f(x)}.$$ Does exist a rule or do you know a way to solve it? $f(x)= (1+x)^n e^{-ax}$ $g(x)= \ln(1+x)$ So: the numerator is the ...
2
votes
0answers
84 views

What do I need to know to integrate any rational function?

My analysis book makes the following statement: Every rational function with real coefficients can be integrated in terms of rational functions, logarithm functions, arctangent ...
3
votes
1answer
272 views

Integrating a partial fraction with multiple quadratic denominators

When integrating a real rational fraction, you first break it into partial fractions. You then end up with fractions with linear denominators $\frac{A}{(x-b)^n}$, which are easy. You also end up with ...
6
votes
4answers
145 views

What goes wrong in this derivative?

$$ f(x) = \frac{2}{3} x (x^2-1)^{-2/3} $$ and f'(x) is searched. So, by applying the product rule $ (uv)' = u'v + uv' $ with $ u=(x^2-1)^{-2/3} $ and $ v=\frac{2}{3} x $, so $ u'=-\frac{4}{3} x ...
3
votes
2answers
237 views

Negative exponents when multiplying polynomials

$$(4-t)(1+t^2)^{-1}$$ I am supposed to find the derivative of this but I am not sure if it means $$\frac{1}{(4-t)(1+t^2)^{-1}}\quad\text{or}\quad\frac{(4-t)}{(1+t^2)}$$ I have tried to look online for ...
2
votes
1answer
76 views

Evaluate a certain derivative

Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $\{l_1,\dots,l_n\}$ a vector of natural numbers such that $l_1+l_2+\dots+l_n=N$. Let $$ h_j(x)=\prod_{i\neq j,i=1,\dots, n} ...
0
votes
2answers
455 views

How to evaluate $\int 1/(1+x^{2n})\,dx$ for an arbitrary positive integer $n$?

How to find $$\int\dfrac{dx}{1+x^{2n}}$$ where $n \in \mathbb N$? Remark When $n=1$, the antiderivative is $\tan^{-1}x+C$. But already with $n=2$ this is something much more complicated. Is there a ...
0
votes
1answer
719 views

How to find the equation of a graph of a rational function from a set of points?

For example, the data points are: (1,1) (2,1/2) (3,1/3) (4,1/4) (5,1/5) How do I find the equation from those points? Do I look at the common ratio of the y-values or something?
1
vote
1answer
175 views

Perturbations of Rational Numbers

For $a_1,a_2$, $b_1,b_2$ $\in\mathbb{R}^+$, if $a_1<b_1$ , then for any perturbation $\epsilon\in \mathbb{R}^+$, $$r_1=\frac{a_1+\epsilon}{b_1+\epsilon}>\frac{a_1}{b_1} $$ and if $a_2>b_2$, ...
2
votes
1answer
229 views

Integration of rational functions..

Some rational function is giving me some trouble... \begin{aligned} \ \int \frac {x^2-9x+16}{(x-1)(x^2+6x-7)} dx \end{aligned} I simplified it like so: \begin{aligned} \ \int \frac ...