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: $\int \frac{\large 2cos\: x-3sin\: x}{\large 6cos\: x+4sin\: x}$

Can you answer this question?
 
 

1 Answer

0 votes
Toolbox:
  • (i)Method of substitution :
  • Given 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 $\frac{dx}{dt}=g'(t).$
  • $\Rightarrow $dx=g'(t)dt.
  • Thus $I=\int f(g(t).g'(t))dt.$
  • (ii)$\int \sin xdx=-\cos x.$
  • (iii)\int\cos xdx=\sin x+c.$
Given $I=\int \frac{2\cos x-3\sin x}{6\cos x+4\sin x}dx.$
 
$\;\;\;\;\qquad=\int \frac{2\cos x-3\sin x}{2(3\cos x+2\sin x)}dx.$
 
Put $3\cos x+2\sin x=t.$
 
$-3\sin x+2\cos x=dt.$
 
Now substituting for x and dx we get,
 
$I=\frac{1}{2}\int \frac{dt}{t} .$
 
On integrating we get,
 
$\frac{1}{2}[log |t|]+c.$
 
Substituting back for t we get,
 
$\int \frac{2\cos x-3\sin x}{6\cos x+4\sin x}dx=\frac{1}{2}log (3\cos x+2\sin x)+c.$
 
 

 

answered Jan 29, 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
...