Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 177775 by depressiveshrek last updated on 08/Oct/22

$$\int \frac{\sin 2x}{\sqrt{{\cos }^{4}x+1}}dx$$

Answered by Ar Brandon last updated on 08/Oct/22

$$I=\int \frac{\sin 2x}{\sqrt{{\cos }^{4}x+1}}dx=\int \frac{2\sin x\cos x}{\sqrt{{\cos }^{4}x+1}}dx,c=\cos x$$$$=-\int \frac{2c}{\sqrt{c^4+1}}dc=-\int \frac{1}{\sqrt{t^2+1}}dt,t=c^2$$$$=-\mathrm{argsh}\left(t\right)+C=-\mathrm{argsh}\left(c^2\right)+C$$$$=-\mathrm{argsh}\left({\cos }^{2}x\right)+C$$$$=-\ln ({\cos }^{2}x+\sqrt{{\cos }^{4}x+1})+C$$$$=\ln \mid \frac{k}{{\cos }^{2}x+\sqrt{{\cos }^{4}x+1}}\mid$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com