Method 1 - #u# - substitution
Lets make a #u# substitution...
let # u = sinx #
then #du = color(red)(cosx dx #
Lets re-write this integral: #int cos^3 x dx -= int cos^2 x * color(red)(cosx dx#
Now we can transform this integral...
# => int cos^2 x * du #
#=> u^2 = sin^2 x #
#=> 1 - u^2 = cos^2 x # #" Using our trig identities..." #
#=> int 1-u^2 du = u - 1/3 u^3 + c " Using reverse power rule..." #
Substituting #u= sinx # back:
#color(blue)(int cos^3 x dx = sinx - 1/3 sin^3 x + c_0 #
#c_0 - "Constant" #
Method 2 - complex numbers
This method invloves the use of complex numbers
We must utilise: #color(orange)( omega + 1 / omega = 2cosx #
Where #omega = cosx + i sinx #
Hence #(omega + 1 / omega) ^3 = 2^3 cos^3 x #
#=> 8cos^3 x = omega^3 + (3omega^2 * 1/omega) + (3omega * 1/omega^2 ) + 1/omega^3 #
#=> 8cos^3 x = omega^3 + 1/omega^3 + 3( omega + 1/omega) #
We know #2cosn x = omega^n + 1/omega^n #
#=> 8cos^3x = 2cos3x + 6cosx #
#=> cos^3x = 1/4(cos3x + 3 cosx) #
Hence our integral becomes: #1/4 int cos3x + 3sinx dx #
#= color(blue)(1/12 sin3x + 3/4 cosx + c_1 #
#c_1 - "Constant" #
Both of these solutions are equal...
#color(red)(sinx - 1/3 sin^3 x + c) or color(blue)(1/12 sin3x + 3/4 cosx + c#: