<s>[INST] To evaluate $\int_0^{\frac{\pi}2} \frac{\cos x}{x+2} dx $ and $\int_0^4{\frac{\sin x \cos x}{(x+1)^2}} dx$ How to evaluate the following integrals?
$$\int_0^{\frac{\pi}2} \frac{\cos x}{x+2} dx $$
$$\int_0^4{\frac{\sin x \cos x}{(x+1)^2}} dx$$
 [/INST] Using what Lucian answered $$\int_0^{\frac{\pi}2} \frac{\cos x}{x+2} dx=\left(\text{Ci}\left(2+\frac{\pi }{2}\right)-\text{Ci}(2)\right) \cos
   (2)+\left(\text{Si}\left(2+\frac{\pi }{2}\right)-\text{Si}(2)\right) \sin (2)$$ 
$$\int_0^4{\frac{\sin x \cos x}{(x+1)^2}} dx=\Big(\text{Ci}(10)-\text{Ci}(2)\Big) \cos (2)-\frac{1}{5} \sin (2) \Big(5 \text{Si}(2)-5
   \text{Si}(10)+\cos (2)+\cos (6)\Big)$$ The numerical values are respectively $0.3971615867$ and $0.1441361714$.
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