logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
Home  >>  CBSE XII  >>  Math  >>  Integrals
0 votes

Integrate the function $x\;\sqrt{x+2}$

Can you answer this question?
 
 

1 Answer

0 votes
Toolbox:
  • Method of substitution:
  • Given $\int f(x)dx$ can be transformed into another form by changing independent variable x to t by substituting x=g(t).
  • Consider $I=\int f(x)dx.$
  • Put x=g(t) so that $\frac{dx}{dx}=g'(t).$
  • dx=g'(t)dt.
  • Thus $ I=\int f(g(t).g'(t))dt.$
Given $ I=x\sqrt {x+2}.$
 
Let t=x+2 $\Rightarrow x=t-2.$
 
Hence dt=dx.
 
Substituting for t and dt we get,
 
$I=\int (t-2)\sqrt tdt.$
 
$\;\;\;=\int(t\sqrt t-2\sqrt t)dt.$
 
$\;\;\;=\int t^\frac{3}{2}dt-2\int t^\frac{1}{2}dt.$
 
On integrating we get,
$\;\;\;=\frac{t^\frac{5}{2}}{\frac{5}{2}}-2\frac{t^\frac{3}{2}}{\frac{3}{2}}+c.$
 
$\;\;\;=\frac{2}{5}t^\frac{5}{2}-\frac{4}{3}t^\frac{3}{2}+c.$
 
Substituting back for t we get,
 
$\;\;\;=\frac{2}{5}(x+2)^\frac{5}{2}-\frac{4}{3}(x+2)^\frac{3}{2}+c.$

 

answered Jan 28, 2013 by sreemathi.v
edited Jan 28, 2013 by sreemathi.v
 
Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...