<s>[INST] Comparison test for $\int_0^{\pi/2} \frac{x+\cos^2 x}{\sqrt{x+x^2}} \,dx$ $$\int_0^{\pi/2} \frac{x+\cos^2 x}{\sqrt{x+x^2}} \,dx$$
How come up with something to compare it to use the comparison test?
 [/INST] In the same spirit as user71352 $$\int_0^{\pi/2} \frac{x+\cos^2 x}{\sqrt{x+x^2}} \,dx \lt \int_0^{\pi/2} \frac{x+1}{\sqrt{x+x^2}} \,dx$$ and $$\int \frac{x+1}{\sqrt{x+x^2}} \,dx=\sqrt{x (x+1)}+\sinh ^{-1}\left(\sqrt{x}\right)$$ $$\int_0^{\pi/2} \frac{x+1}{\sqrt{x+x^2}} \,dx=\frac{1}{2} \left(\sqrt{\pi  (2+\pi )}+\log \left(1+\pi +\sqrt{\pi  (2+\pi
   )}\right)\right) \simeq 3.05919$$ while a numerical integration leads to $$\int_0^{\pi/2} \frac{x+\cos^2 x}{\sqrt{x+x^2}} \,dx \simeq 2.50272$$
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