Question

If $\int \frac{2\cos x-\sin x+\lambda }{\cos x+\sin x-2}dx=A\ln |\cos x+\sin x-2|+Bx+C$, then the value of $\lambda AB$ is____.

Answer 1

Answer: -0.75

$\frac{d}{dx}(A\ln |\cos x+\sin x-2|+Bx+C)=A\frac{\cos x-\sin x}{\cos x+\sin x-2}+B$
$=\frac{A\cos x-A\sin x+B\cos x+B\sin x-2B}{\cos x+\sin x-2}$
$\therefore 2=A+B,-1=-A+B,\lambda =-2B$
$\therefore A=3/2,B=1/2,\lambda =-1$
$\therefore \lambda AB=-3/4$