Recent content by Delusional

I don't really understand why teachers insist on using explicit substitutions for totally banal integrals. The integral \begin{aligned}\int\frac{f'(x)}{2\sqrt{f(x)}}\;{dx} = \sqrt{f(x)}+\mathcal{C}\end{aligned} is one of the most basic standard integrals. It's like integration by parts when...

\begin{aligned}\sum_{k=1}^{n} (n-k+1)k = (n+1)\sum_{k=1}^{n}k-\sum_{k=1}^{n}k^2= (n+1)\left[ \frac{1}{2} n (n+1)\right]-\frac{1}{6}n (n+1) (2 n+1) = \frac{1}{6}n (n+1) (n+2).\end{aligned}