Create an integral whereby you are forced to use all four types of integration. Work the problem and explain why each (u-substitution, trig substitution, fractions, parts) are all needed. this must be only one integral, that is it must all be under a singular fraction and cannot be the sum such as integral of lnx+arctanx dx or something.
Please see the attached file.
∫ (x +1)_tan^2x sec ^2 x___dx
( 1+tan^2 x) (1+3tan^2x)
here tan^2x is tan square x
here sec ^2 x is secant square x
put tan x =z ; sec^2 x dx = dz ( Trignometric substitution )
Substituting the values we get
∫ (tan^-1 z +1)____z^2 _dz__
(1+ z^2) (1+3z^2)
using partial fractions we get
∫ (tan^-1 z +1) [ ____1 ___ - ____1___ ]
2(1+ z^2) 2(1+3z^2)
∫ [_(tan^-1 z +1)_ - __(tan^-1 z +1)__ ]
2(+ z^2 ) 2(1+3z^2)
∫ 1 [_(tan^-1 z ) + 1_ - __tan^-1 z - ___1__ ] dz
2 ( 1+ z^2 ) (1+ z^2 ) (1+3z^2) ...
Examples of integral creations are determined. The singular fractions which cannot be the sum such as an integral lnx+arctanx dx are analyzed.