Question: $$\left[(k+x)\left(2-\frac{x}{2}\right)\right]^6$$

where the coefficient of $x^{2}$ is $84$.Find the value of the constant $k$.
I tried to expand the equation but got a equation of degree 6 for some reason.

Answer: 
**Hint:**
$$\frac{\sin(x+1)\cos(x)}{\sin^2(x)+4} \,dx= \frac{(\sin{x}\cos{1}+\cos{x}\sin{1})\cos(x)}{\sin^2(x)+4} \,dx = \\
 = \cos{1}\frac{\sin{x}\,d(\sin{x})}{\sin^2(x)+4}+ \sin{1}\frac{1-\sin^2{x}}{\sin^2(x)+4}\,dx.$$
Can you proceed?
Hint...start by expanding the $\sin(x+1)$ term using the compound angle identity, so thatvyou have two integrals $$\cos1\int\frac{\sin2x}{9-\cos 2x}dx+\sin1\int\frac{1}{1+4\tan^2x}dx$$

The first is immediately apparent, and the second will work out using $t=\tan x$ and application of partial fractions.