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The partial fraction decomposition isTaking antiderivatives and using the positive endpoints givesHence $\boxed{n=2}$.

Apply implicit differentiation, remembering that $y$ depends on $x$:Since $1+e^y>0$, division is always legitimate. Therefore

The chain rule and the quotient rule giveThe exponential function and denominator are positive, so the stationary points occur exactly at $x=\pm 1$. Their values areThe derivative is negative for $|x|>1$ and positive for $|x|<1$, so $x=-1$ is a local minimum and $x=1$ a local maximum. They are also global extrema, since $-1/2\le x/(1+x^2)\le 1/2$.

This is a finite geometric series with common ratio $r=1/\sqrt{2}$. Since $r^{10}=1/32$,Thus $\boxed{p=31,q=31}$.

The addition formula for the sine givesEquating the sine and cosine coefficients gives $A\cos \varphi =\sqrt{3}$ and $A\sin \varphi =1$. Squaring and adding yields $A^2=4$, so $A=2$ because $A>0$. Both coefficients are positive, so $\varphi$ is in the first quadrant. Consequently

Completing the square transforms the equation intoComparison with the standard circle equation gives

Use the first of the simultaneous equations to put $y=1-x$. Substitution into the second givesThus $x=0$ or $x=3$. The corresponding values of $y$ give exactly

The given polynomial root supplies the factor $2x-3$. Polynomial division givesSolving the remaining quadratic equation, the other two polynomial roots are