Calculus: Early Transcendentals, Chapter 7, 7.2, Section 7.2, Problem 47

int(1-tan^2(x))/(sec^2(x))dx
Rewrite the integrand as,
int(1-tan^2(x))/(sec^2(x))dx=int(1-(sin^2(x))/(cos^2(x)))/(1/(cos^2(x)))dx
=int(cos^2(x)-sin^2(x))dx
Now use the following identities:
cos^2(x)=(1+cos(2x))/2
sin^2(x)=(1-cos(2x))/2
=int((1+cos(2x))/2-(1-cos(2x))/2)dx
=int(1+cos(2x)-1+cos(2x))/2dx
=intcos(2x)dx
=sin(2x)/2
add a constant C to the solution,
=sin(2x)/2+C