Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 147867 by Khalmohmmad last updated on 24/Jul/21

$$\int \sec xdx=?$$

Answered by Olaf_Thorendsen last updated on 24/Jul/21

$$\mathrm{F}\left(x\right)=\int \sec xdx$$$$\mathrm{F}\left(x\right)=\int \frac{dx}{\cos x}$$$$\mathrm{Let}t=\tan \frac{x}{2}$$$$\mathrm{F}\left(t\right)=\int \frac{1}{\frac{1-t^2}{1+t^2}}.\frac{2dt}{1+t^2}$$$$\mathrm{F}\left(t\right)=\int (\frac{1}{1-t}+\frac{1}{1+t})dt$$$$\mathrm{F}\left(t\right)=\ln \mid \frac{1+t}{1-t}\mid +\mathrm{C}$$$$\mathrm{F}\left(x\right)=\ln \mid \frac{1+\tan \frac{x}{2}}{1-\tan \frac{x}{2}}\mid +\mathrm{C}$$$$\mathrm{F}\left(x\right)=\ln \mid \frac{\cos \frac{x}{2}+\sin \frac{x}{2}}{\cos \frac{x}{2}-\sin \frac{x}{2}}\mid +\mathrm{C}$$$$\mathrm{F}\left(x\right)=\ln \mid \frac{\sqrt{2}\cos (\frac{\pi }{4}-\frac{x}{2})}{\sqrt{2}\sin (\frac{\pi }{4}-\frac{x}{2})}\mid +\mathrm{C}$$$$\mathrm{F}\left(x\right)=\ln \mid \cot (\frac{\pi }{4}-\frac{x}{2})\mid +\mathrm{C}$$

Answered by puissant last updated on 24/Jul/21

$$\int \frac{1}{\mathrm{cosx}}\mathrm{dx}$$$$=\int \frac{\mathrm{cosx}}{{\cos }^2\mathrm{x}}\mathrm{dx}$$$$=\int \frac{\mathrm{cosx}}{1-{\sin }^2\mathrm{x}}\mathrm{dx}$$$$\mathrm{let}\mathrm{u}=\mathrm{sinx}\Rightarrow \mathrm{du}=\mathrm{cosx}\mathrm{dx}$$$$=\int \frac{1}{1-{\mathrm{u}}^2}\mathrm{du}$$$$=\frac{1}{2}\int \frac{1}{1+\mathrm{u}}\mathrm{du}+\frac{1}{2}\int \frac{1}{1-\mathrm{u}}\mathrm{du}$$$$=\frac{1}{2}\ln (1+\mathrm{u})-\frac{1}{2}\ln (1-\mathrm{u})+\mathrm{k}$$$$=\frac{1}{2}\ln \left(\frac{1+\mathrm{u}}{1-\mathrm{u}}\right)+\mathrm{k}$$$$\Rightarrow \mathrm{I}=\frac{1}{2}\ln \left(\frac{1+\mathrm{sinx}}{1-\mathrm{sinx}}\right)+\mathrm{k}...$$

Answered by iloveisrael last updated on 24/Jul/21

$$\int \sec \mathrm{x}\left(\frac{\sec \mathrm{x}+\tan \mathrm{x}}{\sec \mathrm{x}+\tan \mathrm{x}}\right)\mathrm{dx}$$$$=\int \frac{\sec {}^2\mathrm{x}+\sec \mathrm{x}\tan \mathrm{x}}{\sec \mathrm{x}+\tan \mathrm{x}}\mathrm{dx}$$$$=\int \frac{\mathrm{d}(\sec \mathrm{x}+\tan \mathrm{x})}{\sec \mathrm{x}+\tan \mathrm{x}}$$$$=\ln \mid \sec \mathrm{x}+\tan \mathrm{x}\mid +\mathrm{c}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com