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  4. If ∫ (2 cos x- sin x+λ/ cos x+ sin x-2) d x=A ln | cos x+ sin x-2|+B x+ C, then the value of λ A B is.

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

Integrals

Solution:

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