Question Number 186181 by cortano1 last updated on 02/Feb/23
$$\:\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\sqrt{\mathrm{1}+\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx}\:=? \\ $$
Answered by MJS_new last updated on 02/Feb/23
![∫(√(1+cos^2 x)) dx=∫(√(2−sin^2 x)) dx= =(√2)∫(√(1−(1/2)sin^2 x)) dx=(√2)E (x∣(1/2)) +C not sure if we can give an exact value... (√2)[E (x∣(1/2))]_0 ^π ≈3.82019778903](https://www.tinkutara.com/question/Q186199.png)
$$\int\sqrt{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \:{x}}\:{dx}=\int\sqrt{\mathrm{2}−\mathrm{sin}^{\mathrm{2}} \:{x}}\:{dx}= \\ $$$$=\sqrt{\mathrm{2}}\int\sqrt{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}^{\mathrm{2}} \:{x}}\:{dx}=\sqrt{\mathrm{2}}\mathrm{E}\:\left({x}\mid\frac{\mathrm{1}}{\mathrm{2}}\right)\:+{C} \\ $$$$\mathrm{not}\:\mathrm{sure}\:\mathrm{if}\:\mathrm{we}\:\mathrm{can}\:\mathrm{give}\:\mathrm{an}\:\mathrm{exact}\:\mathrm{value}… \\ $$$$\sqrt{\mathrm{2}}\left[\mathrm{E}\:\left({x}\mid\frac{\mathrm{1}}{\mathrm{2}}\right)\right]_{\mathrm{0}} ^{\pi} \approx\mathrm{3}.\mathrm{82019778903} \\ $$