How do I evaluiate #intsec(x)(sec(x) + tan(x)) dx#?

1 Answer
May 1, 2018

#int secx(secx+tanx) dx = secx + tanx + c#

Explanation:

I don't believe that there is any trigonometric substitution involved here actually.

The integrand #secx(secx+tanx)# can be expanded out to get #sec^2x +secxtanx# which are actually two simple functions to be antidifferentiated.

#int sec^2x dx = tanx + c# because #d/dx(tanx) = sec^2x#.

#int secxtanx dx = secx# which we will prove using substitution.

Expressing that in terms of sine and cosine, we get #secxtanx = sinx/cos^2x# so the integral becomes:

#int sinx/cos^2x dx#, to which we apply the substitution #u=cosx#.

#u=cosx#
#:. (du)/dx=-sinx#
#:. -du = sinxdx#

Now, using the change of variable rule, we get:

# - int 1/u^2 du#
#=1/u + c#
#= secx + c#

#:. int secx(secx+tanx) dx = secx + tanx + c#

And there you have it!