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Find the integral $\int(1-x)\sqrt x\;dx$

$\begin{array}{1 1} \large \frac{2}{3}x^\frac{3}{2}-\large\frac{2}{5}x^\frac{5}{2}+c. \\ \large \frac{2}{3}x^\frac{-3}{2}-\large\frac{-2}{5}x^\frac{5}{2}+c. \\ \large \frac{2}{3}x^\frac{-3}{2}+\large\frac{2}{5}x^\frac{5}{2}+c. \\ \large \frac{2}{3}x^\frac{3}{2}+\large\frac{2}{5}x^\frac{5}{2}+c. \end{array} $

1 Answer

Toolbox:
  • $(i)\int x^n dx=\frac{x^{n+1}}{n+1}+c$.
$\int(1-x)\sqrt xdx.$
 
On expanding we get,
 
$\int(\sqrt x-x^\frac{3}{2})dx$.
 
$\;\;\;=\int x^\frac{1}{2}-\int x^\frac{3}{2}dx.$
 
$\;\;\;=\frac{x^\frac{1}{2}+1}{\frac{1}{2}+1}-\frac{x^\frac{3}{2}+1}{\frac{3}{2}+1}+c.$
 
$\;\;\;=\frac{2}{3}x^\frac{3}{2}-\frac{2}{5}x^\frac{5}{2}+c.$
 
Hence $\int(1-x)\sqrt xdx=\frac{2}{3}x^\frac{3}{2}-\frac{2}{5}x^\frac{5}{2}+c.$

 

answered Jan 27, 2013 by sreemathi.v
 
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