CDS-1 2012 ANSWER KEY WITH SOLUTIONS OF ELEMENTARY MATHEMATICS

CDS-1 2012 ANSWER KEY WITH SOLUTIONS OF ELEMENTARY MATHEMATICS

New invented formulae based on INTEGRATION

Dear friends,
This is a newly invented formula and it is mainly useful for students who are studying in 12th standard and the main use of this is to just understand that how a complex integration can we solved.
Here is the procedure to understand and learn the newly invented formulae based on integration.

Name of the formulae :

 ∫e^ax tanbx dx 


Procedure / proof:-


Let D=e^ax sinbx → 1


          E=e^ax cosbx→2
differentiating 1 & 2 with respect to x on both sides so we get,


0 = e^axbcosbx + ae^axsinbx → 3


0 = e^ax-bsinbx + ae^axcosbx → 4


Now add 3+4 so we get the equation as follows,


0 = e^axbcosbx + ae^axsinbx + e^ax-bsinbx + ae^axcosbx


now dividing on both side with cos bx


0 = e^axbcosbx  + ae^axsinbx  + e^ax - bsinbx + ae^axcosbx


0 = be^ax + ae^axtanbx - be^axtanbx + ae^ax


0 = (a-b)e^axtanbx + (a+b)e^ax


Now integrating on both sides with respect to dx


 ∫ 0dx = (a-b)  ∫e^axtanbx dx + (a+b) ∫e^ax dx


-(a+b)e^ax/a=(a-b) ∫e^axtanbx dx


-(a+b)e^ax/a * 1/(a-b) = ∫e^axtanbx dx


so dear friends the main application of this formula will be useful for the understanding of the problem of type ∫e^ax tanbx dx 




example 1: Now let us illustrate this formula with some problems.


∫e^2x tanx dx 


in the above problem a=2,b=1 so the solution is directly obtained as


(-1) * e^2x/2 * 1/1= -e^2x/2


ans = -e^2x/2


If you have any problem regarding this then contact me on sharadsuman.jha95@gmail.com