Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Model Papers
0 votes

Evaluate : $ \int \large\frac{6x+7}{\sqrt{(x-5)(x-4)}}$$dx $

Can you answer this question?

1 Answer

0 votes
  • $\int\large\frac{(px+q)}{\sqrt{ax^2+bx+c}}$$dx.$,where p,q,a,b,c are constants,we are to find real numbers such that $px+q=A\large\frac{d}{dx}$$(ax^2+bx+c)+B$
  • Therefore $px+q=A(2ax+b)+B.$
  • To determine A and B we equate the coefficients from both sided,the coefficients of x and the constant terms.A and B are thus obtained
  • $\int\large\frac{dx}{\sqrt{x^2-a^2}}$$=\log|x+\sqrt{x^2-a^2}|+c.$
Step 1:
$I=\int\large \frac{6x+7}{(x-5)(x-4)}$$dx=\int\large\frac{6x+7}{\sqrt{x^2-9x+20}}.$
We can write,
Now equating the coefficients we get,
6=2A$\Rightarrow A=3.$
$7=-9A+B\Rightarrow B=34.$
Hence A=3 and B=34.
Step 2:
On separating we can write,
Let $x^2-9x+20=t.$ and $x^2-9x+20=(x-\frac{9}{2})^2-\frac{1}{4}=(x-\large\frac{9}{2})^2-(\frac{1}{2})^2$.
On differentiating
Therefore $I=3\int\frac{dt}{\sqrt t}+34\int\frac{dx}{\sqrt{(x-\frac{9}{2}}})^2-(\large\frac{1}{2})^2.$
Step 3:
On integrating we get,
$\;\;\;=3\times 2(\sqrt t)+34\log|(x-9/2)+\sqrt{(x-5)(x-4)}+c.$
Substituting for t we get,
answered Nov 11, 2013 by sreemathi.v
Ask Question
student study plans
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App