PART 16：避重就輕原則

當遇到此類題型 \(\int {x{{(1 - x)}^{20}}\;dx} \)

我們不希望20次方位置放在 \((1 - x)\) 之上，

所以令 \(u = 1 - x\;\;\)  \( \Rightarrow \;\;x = 1 - u\;\;\)  \(\; \Rightarrow \;\;dx =  - du\)

SOL:

原式 = \(\int {x{{(1 - x)}^{20}}\;dx}  \)   \(= - \int {(1 - u){u^{20}}\;du} \)  

\( = \int {(u - 1){u^{20}}\;du} \)   \( = \int {({u^{21}} - {u^{20}}\;)du} \)  

\( = \frac{{{u^{22}}}}{{22}} - \frac{{{u^{21}}}}{{21}} + C\)   \( = \frac{{{{(1 - x)}^{22}}}}{{22}} - \frac{{{{(1 - x)}^{21}}}}{{21}} + C\)