Why does free imply torsion-free?
I want to verify that if $R$ is an integral domain and $M$ is an $R$-module, that if $M$ is free, $M$ must also be torsion-free. Where can I start with th...
Apr 19, 2026 1 VIEWS
Topic: General
5104 Articles 
Integrate $\int \ln(x^2 +1)\ dx$
$$\int \ln(x^2 +1)\ dx$$ I done it using integration by parts where $\int u\ dv = uv - \int v\ du$ Let $u$ = $\ln(x^2 +1)$ $du = \frac{2x}{x^2+1} dx $ Let...
Apr 19, 2026 1 VIEWS
Given a $17^\circ$ angle, construct a $1^\circ$ angle using only a compass and straightedge
Given a $17^\circ$ angle, construct a $1^\circ$ angle using only a compass and straightedge I've tried creating 20 seventeen angles so that there is ...
Apr 19, 2026 1 VIEWS
Can someone easily explain Artin's conjecture on primitive roots?
Wikipedia (1): In number theory, Artin's conjecture on primitive roots states that a given integer a that is neither a perfect square nor −1 is a pri...
Apr 19, 2026 3 VIEWS
Examples of methods for solving modular equations
Simple mod questions. Can you show example to do such things? $x+40 \equiv 1 \pmod{88}$. $x \cdot 40 \equiv 1 \pmod{88}$. $5a+3b \equiv 1 \pmod{11}$ and $...
Apr 19, 2026 3 VIEWS
Truth table confusion
I'm given a conditional statement $ ~p \lor q \rightarrow r$ From what I understand, the state of $r$ is dependent on the state of $p$ and $q$. The t...
Apr 19, 2026 1 VIEWS
Determine the Taylor series for $f(x) = x^3 \cdot \ln{\sqrt{x}}$ around the point $a = 1$ and determine its radius of convergence.
As the question states: "Determine the Taylor series for $f(x) = x^3 \cdot \ln{\sqrt{x}}$ around the point $a = 1$ and determine its radius of convergence...
Apr 19, 2026 1 VIEWS
Modus Ponens Proof
I have written the truth table for all of the forms of $P$ and $Q$.Then maintained the table to find $P \rightarrow Q $ and $[(P \rightarrow Q) \wedge P]$...
Apr 19, 2026 1 VIEWS
LIATE / ILATE rule
In another question of mine users proposed the LIATE or ILATE rule for partial integration. However, I have encountered a problem: $$ e^{-x}\cos(x)$$ If I...
Apr 19, 2026 1 VIEWS
How do you deal with inverse trig functions that produce results outside their domain?
The problem I am facing is this: Find $\cos(X) = 4/7$ in quadrant IV (of the Cartesian plane) The next step leads me to this: $X = \cos^{-1}(4/7)$ However...
Apr 19, 2026 1 VIEWS
calculation of median of grouped data
While calculating the median of grouped data of total frequency $N$, in order to find the median class which value should be taken into consideration to m...
Apr 19, 2026 1 VIEWS
What is the theory of Matrices?
I've just passed high school and studying matrices. I've learned about determinant, transpose and adjoint etc. I've learned the method of f...
Apr 19, 2026 2 VIEWS