Answer to Question #167812 in Calculus for Phyroe

Question

Evaluate the integral of (dt)/(cos² t)

Answer 1

$\int\frac{dt}{\cos^2{\ \left(t\right)}}$

 $let\ x=\tan{\left(t\right)} → \cos{\left(t\right)}=\frac{1}{\sqrt{1+x^2}},\sin{\left(t\right)}=\frac{x}{\sqrt{1+x^2}}\$

$dx=\sec^2{\left(t\right)\ dt}$

$dt=\cos^2{\left(t\right)\ dx\ }$

$dt=\frac{dx}{1+x^2}$

$\int\frac{dt}{\cos^2{\left(t\right)}}=\int\frac{1+x^2}{1}\frac{dx}{1+x^2}\ =\ \int dx$

$x+c$

$\tan{\left(t\right)}+c$