How do you find #int sec^2x/(1-sin^2x) dx #?

2 Answers
May 26, 2018

Shown below

Explanation:

#sin^2 x + cos^2 x = 1 => cos^2x = 1 - sin^2 x #

#=> int sec^2x / ( cos^2 x ) dx #

#=> int sec^2 x * 1/cos^2 x dx #

#=> int sec^4 x dx #

#=> int sec^2 x * sec^2 x dx #

#=> int ( 1 + tan^2x) * sec^2 x dx #

# u = tanx #

#du = sec^2 x dx #

#=> int 1 + u ^2 du #

#=> u + 1/3 u^3 + c #

# = tanx + 1/3 tan^3 x + c #

May 26, 2018

#tan(x) + frac(1)(3) tan^(3)(x) + C#

Explanation:

We have: #int# #frac(sec^(2)(x))(1 - sin^(2)(x))# #dx#

#= int# #frac(sec^(2)(x))(cos^(2)(x))# #dx#

#= int# #frac(sec^(2)(x))(frac(1)(sec^(2)(x)))# #dx#

#= int# #sec^(2)(x) cdot sec^(2)(x)# #dx#

Then, the Pythagorean identity is #cos^(2)(x) + sin^(2)(x) = 1#.

We can divide through by #cos^(2)(x)# it to get:

#Rightarrow 1 + tan^(2)(x) = sec^(2)(x)#

Let's apply this rearranged identity to our integral:

#= int# #(1 + tan^(2)(x)) cdot sec^(2)(x)# #dx#

Now, let's use #u#-substitution, where #u = tan(x) Rightarrow du = sec^(2)(x)# #dx#:

#= int# #(1 + u^(2))# #du#

#= int# #1# #du + int# #u^(2)# #du#

#= u + frac(1)(3) u^(3) + C#

Finally, we can substitute #tan(x)# in place of #u#:

#= tan(x) + frac(1)(3) tan^(3)(x) + C#