Let S=(0,2)-2,34,32,74.

Question

Let $S=(0,2\pi)-\left\{\dfrac{\pi}{2},\dfrac{3\pi}{4},\dfrac{3\pi}{2},\dfrac{7\pi}{4}\right\}$. Let $y=y(x)$, $x\in S$, solve $\dfrac{dy}{dx}=\dfrac{1}{1+\sin2x}$, $y\left(\dfrac{\pi}{4}\right)=\dfrac12$. If the sum of abscissas of intersection of $y=y(x)$ with $y=\sqrt2\sin x$ is $\dfrac{k\pi}{12}$, then $k$ is \_\_\_\_\_\_.

Answer: 42

Accepted Answer

Integrating $\dfrac{1}{1+\sin2x}=\dfrac{1}{(\sin x+\cos x)^2}$ gives $y=\dfrac{	an x-1}{	an x+1}$-type expression after using $y(\pi/4)=1/2$; solving $y(x)=\sqrt2\sin x$ over $S$ and summing the roots gives $k=42$.

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