How to evaluate $$I=\int_{0}^{\frac{\pi}{2}}\frac{\sin (x+\sin x)}{\sin x}\,dx$$ I tried to expand $$I=\int_{0}^{\frac{\pi}{2}}\frac{\sin x\cos (\sin x)+\cos x\sin (\sin x)}{\sin x}\,dx$$ $$=\int_{0}^{\frac{\pi}{2}}\left(\cos (\sin x)+\frac{\cos x\sin (\sin x)}{\sin x}\right)dx$$ $$=\int_{0}^{\frac{\pi}{2}}\cos (\sin x)\,dx+\int_{0}^{\frac{\pi}{2}}\frac{\sin (\sin x)}{\sin x}\,d(\sin x)$$ $$=\int_{0}^{\frac{\pi}{2}}\cos (\sin x)\,dx+\int_{0}^{1}\frac{\sin x}{x}\,dx$$ $$=\int_{0}^{\frac{\pi}{2}}\cos (\sin x)\,dx+\operatorname{Si}(1)$$ But I don't know what to do next, help me, thank you
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1$\begingroup$ Check bessel's function. $\endgroup$ – Ajay Mishra Jun 20 at 5:25
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$\begingroup$ Thank you very much! $\endgroup$ – FofX Jun 20 at 8:01
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What you did is correct! Let me help you out a little bit. The bessel function's integral representation is given by $$J_n(z)=\frac1\pi \int\limits_{0}^{\pi}\cos(tn-z\sin t) dt$$ Thus, $$J_0(1)=\frac1\pi \int\limits_{0}^{\pi}\cos(-\sin t) dt$$ $$J_0(1)=\frac2\pi \int\limits_{0}^{\pi/2}\cos(\sin t) dt$$
which gives your result by replacing variables $$\int\limits_{0}^{\pi/2}\cos(\sin x) dx = \frac{\pi}{2}J_0(1)$$
So, the final asnwer of your integral becomes $$\boxed{\frac{\pi}{2}J_0(1)+ \text{Si}(1)}$$
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1$\begingroup$ I was curious so I looked up the value: $2.148\,053$. The link also has some fun alternate representations. $\endgroup$ – let's have a breakdown Jun 20 at 6:17
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