Integration Questions

Question Number 122542 by bramlexs22 last updated on 18/Nov/20

$$\:\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:\sqrt{\mathrm{1}+\mathrm{cos}\:\left({x}^{\mathrm{2}} \right)}\:{dx}\:? \\$$

Answered by Olaf last updated on 18/Nov/20

$$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{\mathrm{1}+\mathrm{cos}\left({x}^{\mathrm{2}} \right)}{dx} \\$$$$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{\mathrm{2cos}^{\mathrm{2}} \left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)}{dx} \\$$$$\mathrm{I}\:=\:\sqrt{\mathrm{2}}\int_{\mathrm{0}} ^{\mathrm{3}} \mid\mathrm{cos}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)\mid{dx} \\$$$$\mathrm{I}\:=\:\sqrt{\mathrm{2}}\int_{\mathrm{0}} ^{\sqrt{\pi}} \mid\mathrm{cos}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)\mid{dx}+\sqrt{\mathrm{2}}\int_{\sqrt{\pi}} ^{\mathrm{3}} \mid\mathrm{cos}^{\mathrm{2}} \left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)\mid{dx} \\$$$$\mathrm{I}\:=\:\sqrt{\mathrm{2}}\int_{\mathrm{0}} ^{\sqrt{\pi}} \mathrm{cos}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right){dx}−\sqrt{\mathrm{2}}\int_{\sqrt{\pi}} ^{\mathrm{3}} \mathrm{cos}^{\mathrm{2}} \left(\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right){dx} \\$$$$\mathrm{I}\:=\:\sqrt{\mathrm{2}}\left[\sqrt{\pi}\mathrm{FresnelC}\left(\frac{{x}}{\pi}\right)\right]_{\mathrm{0}} ^{\sqrt{\pi}} −\sqrt{\mathrm{2}}\left[\sqrt{\pi}\mathrm{FresnelC}\left(\frac{{x}}{\pi}\right)\right]_{\sqrt{\pi}} ^{\mathrm{3}} \\$$$$\mathrm{I}\:=\:\sqrt{\mathrm{2}\pi}\left[\mathrm{2FresnelC}\left(\frac{\mathrm{1}}{\:\sqrt{\pi}}\right)−\mathrm{FresnelC}\left(\mathrm{0}\right)−\mathrm{FresnelC}\left(\mathrm{3}\right)\right] \\$$$$\mathrm{I}\:=\:\sqrt{\mathrm{2}\pi}\left[\mathrm{2FresnelC}\left(\frac{\mathrm{1}}{\:\sqrt{\pi}}\right)−\mathrm{FresnelC}\left(\mathrm{3}\right)\right] \\$$$$\mathrm{Note}\:: \\$$$$\mathrm{Fresnel}\:\mathrm{Cosine}\:\mathrm{Integral} \\$$$$\mathrm{FresnelC}\left({z}\right)\:=\:\int_{\mathrm{0}} ^{{z}} \mathrm{cos}\left(\frac{\pi{t}^{\mathrm{2}} }{\mathrm{2}}\right){dt} \\$$