Prove

Solution:

$\frac{6 x+7}{\sqrt{(x-5)(x-4)}}=\frac{6 x+7}{\sqrt{x^{2}-9 x+20}}$

Let $6 x+7=A \frac{d}{d x}\left(x^{2}-9 x+20\right)+B$

$\Rightarrow 6 x+7=A(2 x-9)+B$

Equating the coefficients of x and constant term, we obtain

$2 A=6 \Rightarrow A=3$

$-9 A+B=7 \Rightarrow B=34$

$\therefore 6 x+7=3(2 x-9)+34$

$\int \frac{6 x+7}{\sqrt{x^{2}-9 x+20}}=\int \frac{3(2 x-9)+34}{\sqrt{x^{2}-9 x+20}} d x$

$=3 \int \frac{2 x-9}{\sqrt{x^{2}-9 x+20}} d x+34 \int \frac{1}{\sqrt{x^{2}-9 x+20}} d x$

Let $I_{1}=\int \frac{2 x-9}{\sqrt{x^{2}-9 x+20}} d x$ and $I_{2}=\int \frac{1}{\sqrt{x^{2}-9 x+20}} d x$

Let $I_{1}=\int \frac{2 x-9}{\sqrt{x^{2}-9 x+20}} d x$ and $I_{2}=\int \frac{1}{\sqrt{x^{2}-9 x+20}} d x$

$\therefore \int \frac{6 x+7}{\sqrt{x^{2}-9 x+20}}=3 I_{1}+34 I_{2}$      ...(1)

Then,

$I_{1}=\int \frac{2 x-9}{\sqrt{x^{2}-9 x+20}} d x$

Let $x^{2}-9 x+20=t$

$\Rightarrow(2 x-9) d x=d t$

$\Rightarrow I_{1}=\frac{d t}{\sqrt{t}}$

$I_{1}=2 \sqrt{t}$

$I_{1}=2 \sqrt{x^{2}-9 x+20}$

and $I_{2}=\int \frac{1}{\sqrt{x^{2}-9 x+20}} d x$    

and $I_{2}=\int \frac{1}{\sqrt{x^{2}-9 x+20}} d x$

$x^{2}-9 x+20$ can be written as $x^{2}-9 x+20+\frac{81}{4}-\frac{81}{4}$

Therefore,

$x^{2}-9 x+20+\frac{81}{4}-\frac{81}{4}$

$=\left(x-\frac{9}{2}\right)^{2}-\frac{1}{4}$

$=\left(x-\frac{9}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}$

$\Rightarrow I_{2}=\int \frac{1}{\sqrt{\left(x-\frac{9}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}} d x$

$I_{2}=\log \left|\left(x-\frac{9}{2}\right)+\sqrt{x^{2}-9 x+20}\right|$    

$x^{2}-9 x+20$ can be written as $x^{2}-9 x+20+\frac{81}{4}-\frac{81}{4}$

Therefore,

$x^{2}-9 x+20+\frac{81}{4}-\frac{81}{4}$

$=\left(x-\frac{9}{2}\right)^{2}-\frac{1}{4}$

$\Rightarrow I_{2}=\int \frac{1}{\sqrt{\left(x-\frac{9}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}} d x$

$I_{2}=\log \left|\left(x-\frac{9}{2}\right)+\sqrt{x^{2}-9 x+20}\right|$   ...(3)

Substituting equations (2) and (3) in (1), we obtain

$\int \frac{6 x+7}{\sqrt{x^{2}-9 x+20}} d x=3\left[2 \sqrt{x^{2}-9 x+20}\right]+34 \log \left[\left(x-\frac{9}{2}\right)+\sqrt{x^{2}-9 x+20}\right]+C$

$=6 \sqrt{x^{2}-9 x+20}+34 \log \left[\left(x-\frac{9}{2}\right)+\sqrt{x^{2}-9 x+20}\right]+\mathrm{C}$