Evaluate-1-4-x-1-x-5-dx-

Question Number 159938 by ZiYangLee last updated on 22/Nov/21

Evaluate \int_{1}^{4} \sqrt{\frac{x - 1}{x^{5}}} d x .

Answered by Ar Brandon last updated on 22/Nov/21

I = \int_{1}^{4} \sqrt{\frac{x - 1}{x^{5}}} d x = \int_{1}^{4} \frac{1}{x^{2}} \sqrt{1 - \frac{1}{x}} d x

u = 1 - \frac{1}{x} \Rightarrow d u = \frac{1}{x^{2}} d x

I = \int_{0}^{\frac{3}{4}} \sqrt{u} d u = \frac{2}{3} {[u^{\frac{3}{2}}]}_{0}^{\frac{3}{4}} = \frac{2}{3} {(\frac{\sqrt{3}}{2})}^{3} = \frac{\sqrt{3}}{4}