Question 11(c)

Using the substitution $u = x - 1$, find $\displaystyle \int x \sqrt{x - 1} dx$

Solution

This is an example of performing integration by substitution. To solve this, we can apply The Change of Variables Theorem

$\int g(u(x)) \cdot u'(x) dx = \int g(y) dy \quad \text{where } y = u(x)$

where

$$\begin{aligned} u(x) &= x - 1 \\ g(u(x))u'(x) &= x \sqrt{x - 1} \\ \end{aligned}$$

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