Question

Hi! I just haven't get why the answer to the minus integral of xdx is equal to -x^2/2?
I mean, why do we have to divide by 2? Didn't quite get that on the previous lecture.
Thanks.

Answer 1

Let's say
\[ y = - \frac{1}{2} x^2\]
now take the derivative of both sides with respect to x:
\[ \frac{dy}{dx}= -\frac{1}{2} 2 x \\ dy = - -\frac{1}{2} 2 x\ dx \\
dy= -x \ dx \]

the idea is the derivative and the integral are inverse operations (up to a constant)
thus the rule for the integral of x to some power:
\[ \int x^n \ dx= \frac{1}{n+1} x^{n+1} + C \]
and taking the derivative:
\[ \frac{d}{dx} \int x^n \ dx= \frac{d}{dx} \left(\frac{1}{n+1} x^{n+1} + C\right)\\
= \frac{1}{n+1} \frac{d}{dx} x^{n+1} + \cancel{\frac{d}{dx}C}0\\
= \frac{1}{n+1} (n+1) x^{n+1-1} \\
=x^n
\]