An integral misses tangent

Calculus Level 4

0π2sin2x3+2sinx+2cosxdx=arctan1k\large \int_{0}^{\frac{\pi}{2}}\frac{ \sin^2x}{3+2 \sin x+2 \cos x} dx = \arctan \frac{1}{k}

Find the value of kk satisfying the equation above.

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