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  4. If ∫ ( cos x- sin x/√8- sin 2 x) d x=a sin -1(( sin x+ cos x/b))+c where c is a constant of integration, then the ordered pair (a, b) is equal to

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Q. If $\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x=a \sin ^{-1}\left(\frac{\sin x+\cos x}{b}\right)+c$ where $c$ is a constant of integration, then the ordered pair $(a, b)$ is equal to

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Solution:

$\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x$
$=\int \frac{\cos x-\sin x}{\sqrt{9-(\sin x+\cos x)^{2}}} d x$
Let $\sin x+\cos x=t$
$\int \frac{d t}{\sqrt{9-t^{2}}}=\sin ^{-1} \frac{t}{3}+c$
$=\sin ^{-1}\left(\frac{\sin x+\cos x}{3}\right)+c$
So $a=1, b=3$