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  4. If ∫( sin 4 x- cos 4 x) d x=-( sin 2 x/k)+c then find k

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Q. If $\int\left(\sin ^{4} x-\cos ^{4} x\right) d x=-\frac{\sin 2 x}{k}+c$ then find $k$

Integrals

Solution:

$\int\left(\sin ^{4} x-\cos ^{4} x\right) d x$
$=\int\left(\sin ^{2} x-\cos ^{2} x\right)\left(\sin ^{2} x+\cos ^{2} x\right) d x$
$=\int\left(\sin ^{2} x-\cos ^{2} x\right) d x$
$=-\int\left(\cos ^{2} x-\sin ^{2} x\right) d x=-\int \cos 2 x d x$
$=\frac{-\sin 2 x}{2}+c$