What is #int(sinx)/(cos^2 x) dx#?

2 Answers
Apr 21, 2018

#secx+C#

Explanation:

#intsinx/(cos^2(x))dx#

using u-substitution:
let #u=cosx#. then #du=-sinxdx# or #-du=sinxdx#

substituting into the original integral:
#int-1/u^2du#

integrate that with power rule:

#1/u+C#

substitute #u=cosx#:

#1/cosx+C=secx+C#

Apr 21, 2018

#secx+C#

Explanation:

Based on the "inverse" nature of derivatives and antiderivatives (found through integrals), if we know that #d/dxsecx=secxtanx#, then #intsecxtanxdx=secx+C#.

Here, using #1/cosx=secx# and #tanx=sinx/cosx#, we see that

#intsinx/cos^2xdx=int1/cosx(sinx/cosx)dx=intsecxtanxdx=secx+C#