Is $\sqrt{x^2} = x$? The many different modulus functions.

What is $\sqrt{x^2}$? Most of us will intuitive say "$x$": after all, $\sqrt{9} = \sqrt{3^2} = \sqrt{3}$, for example. However, what is $\sqrt{(-3)^2}$?

It is not $-3$ and is in fact $\sqrt{(-3)^2} = \sqrt{9} = 3$. Hence $\sqrt{x^2} =x$ is only valid if $x$ is non-negative. If $x$ is negative, it turns out that $\sqrt{x^2} = -x$.

The reason for this stems from definition: the symbol $\sqrt{ \cdot}$ is defined to be the "positive square root" when there are actually two possible square roots to every positive real number (this is the reason the equation $x^2 = k$ has two solutions, $\pm \sqrt{k}$, for positive $k$).

A compact way to summarize:

Integration techniques

1. Equations and inequalities
2. Curves and transformations
3. Functions
4. Arithmetic and geometric progressions
5. Series and the sigma notation
6. Differentiation
7. Maclaurin series and binomial expansion
8. Integration techniques
9. Definite integrals
10. Differential equations
11. Vectors
12. Complex numbers

Next up: randomly generated question on integration to test your concepts. Then vectors, which will likely take a whole chunk of time and effort. Let’s work hard together!