What is the Integral of dx/ (1+cos ^2(x)) Using Different Approaches?

[belleamie]

for the life of me i can't seem to understand how to the the intergral of dx/ (1+cos ^2(x))?

 

[dextercioby]

Use a substitution

$$\tan\frac{x}{2}=t$$

and some trigonometry.

Daniel.

 

[arildno]

Daniel's approach probably works equally well; here's another approach:
$$\frac{1}{1+\cos^{2}x}=\frac{1}{\cos^{2}x}\frac{1}{1+\frac{1}{\cos^{2}x}}=(\frac{d}{dx}tan(x))\frac{1}{2+\tan^{2}x}$$
Thus, setting $$u=tan(x)$$, we have $$\frac{du}{dx}dx=du$$, that is:
$$\int\frac{dx}{1+\cos^{2}x}=\int\frac{du}{2+u^{2}}$$