Question

If $\int \frac{6x+7}{\sqrt{\left(x-5\right)\left(x-4\right)}}dx$
$=A\sqrt{x^{2}-9x+20}+B\,log\left|x+\sqrt{x^{2}-9x+20}-\frac{9}{2}\right|+C$
Then the values of A and B are

• A. $6, 34$
• B. $3, 9$
• C. $12, 17$
• D. None of these

Answer 1

Answer: (A)

Let $6x+7=\lambda \frac{d}{dx}\left(x-5\right)\left(x-4\right)+\mu$
i.e. $6x+7=\lambda\left(2x-9\right)+\mu$ which gives $\lambda=3$ and $\mu=34$
$\therefore \int \frac{6x+7}{\sqrt{\left(x-5\right)\left(x-4\right)}}dx=\int\frac{3\left(2x-9\right)+34}{x^{2}-9x+20}dx$
$=3\int\left(2x-9\right)\left(x^{2}-9x+20\right)^{-\frac{1}{2}}dx+34\int\frac{dx}{\sqrt{x^{2}-9x+20}}$
$=3\int\frac{\left(x^{2}-9x+20\right)^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+34\int\frac{dx}{\sqrt{x^{2}-9x+\frac{81}{4}-\frac{1}{4}}}$
$=6\sqrt{x^{2}-9x+20}+34\int \frac{dx}{\sqrt{\left(x-\frac{9}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}}$
$=6\sqrt{x^{2}-9x+20}+34\,log\left\{\left|x-\frac{9}{2}+\sqrt{\left(x-\frac{9}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}\right|\right\}+C$
$=6\sqrt{x^{2}-9x+20}+34\,log\left|x+\sqrt{x^{2}-9x+20}-\frac{9}{2}\right|+C$
$\therefore A = 6, B = 34.$