Q. 9 - Calculus

Question

Explain why choosing $u=1$ (and thus choosing dv to be the entire integrand, including dx) is never a good choice for integration by parts. 

Step-by-Step Solution

Answer

Hence proved.

Step 1: Explain the problem with u=1

In integration by parts, \(\int u\,dv = uv - \int v\,du\). If \(u = 1\), then \(du = 0\), and \(dv\) is the entire integrand.

Step 2: Show why it fails

With \(u = 1\) and \(du = 0\): \(\int 1 \cdot dv = 1 \cdot v - \int v \cdot 0 = v\). This means we need to find \(v = \int dv\), which is the original integral. We end up right back where we started, making no progress at all.