What is the integral of #int x/ cos^2(x)dx#?

1 Answer
May 29, 2018

#intx/(cos²(x))dx=xtan(x)+ln(|cos(x)|)+C#, #C in RR#

Explanation:

#intx/(cos²(x))dx=intx*1/(cos²(x))dx#
Using integration by parts :
#f(x)=x#
#f'(x)=1#
#g'(x)=1/(cos²x)#
#g(x)=tan(x)#
So: #intx/(cos²(x))dx=xtan(x)-inttan(x)dx#
#=xtan(x)-intsin(x)/cos(x)dx#
Let #X=cos(x)#
#dX=-sin(x)dx#
So:
#intx/(cos²(x))dx=xtan(x)+int1/XdX#
#intx/(cos²(x))dx=xtan(x)+ln(|X|)#
#intx/(cos²(x))dx=xtan(x)+ln(|cos(x)|)+C#, #C in RR#
\0/ here's our answer!