Evaluate $\int_{\pi/4}^{\pi/2} \frac{3+5cot(x)}{5-3cot(x)}$ I'm asked to evaluate $$\int_{\pi/4}^{\pi/2} \frac{3+5\cot(x)}{5-3\cot(x)}$$. 
If this were just an integral for $\displaystyle\frac{5\cot(x)}{-3\cot(x)}$, I would be able to solve it just fine, but I'm a bit stuck on this one. I can't use u-substitution (or if I can, I don't see how) because there's no relation between the numerator and denominator. I also can't split the fraction in two because those constants being added to the cotangent terms keep getting in the way.
Could someone please offer a hint as to how I should get started with this problem?

Notice that 
\begin{align} \int \frac{5 \cot(x) + 3 }{5-3 \cot(x) } ~\text{d}x &= \int \frac{5 \cot(x) + 3 }{5-3 \cot(x) } \cdot \frac{\tan(x) \frac{1}{\cos(x)^2}}{\tan(x) \frac{1}{\cos(x)^2}}~\text{d}x \\
&=\int \frac{(5+3 \tan(x)) \frac{1}{\cos(x)^2}}{(5\tan(x)-3)(\tan(x)^2+1)}~\text{d}x \\
&=\int \frac{5+3y}{(5y+3)(y^2+1)} ~\text{d}y
\end{align}
where we used that $\tan(x)^2+1=\frac{1}{\cos(x)^2}$ and now use partial fraction decomposition

 $$\frac{5+3y}{(5y+3)(y^2+1)}=\frac{5}{5y-3}-\frac{y}{y^2+1}$$

to evaluate this integral.
