Calculate
$$\int_{\pi/4}^{\pi/2}\frac{2\cos\left(x\right) + \sin\left(x\right) + 1\,}{\sin\left(x\right) + \cos\left(x\right)}\,{\rm d}x
$$
I tried to use substitution $t = \tan\left(x/2\right)$ and I got
$$
\int_{\tan\left(\pi/8\right)}^{1}\,\,\frac{2\left(-t^{2} + 2t + 3\right)}{\left(-t^{2} + 2t + 1\right)\left(1 + t^{2}\right)}\,{\rm d}t
$$
It even seem harder. Anyone can help me solve or give me a hint about this problem $?$. Thank you
Best Answer
Decompose the integrand ad follows
\begin{align} &\frac{2\cos{x} + \sin{x}+1}{\sin{x}+\cos{x}}\\ =& \ \frac{\frac32(\cos{x} + \sin{x})+\frac12 (\cos{x} - \sin{x}) +1}{\sin{x}+\cos{x}}\\ =&\ \frac32+\frac12 \frac{(\sin{x} +\cos{x})’}{\sin x+\cos x}+ \frac{(\sin{x} -\cos{x})’}{2-(\sin{x}-\cos{x})^2} \end{align}