Integration by Tabular Method

Topic : Integration by Tabular Method
Question : Find ∫x³ Cos 2x dx


Hints :
∫x^n Cos ax dx
Take u = x^n
dv = Cos ax dx

For patterns like
∫x^n Sin ax dx , ∫x^n Cos ax dx ,
∫x^n e^(ax) dx we use tabular method


Solution :

∫x³ Cos 2x dx
u = x³ and dv = Cos 2x











+x³(Sin 2x)/2 - 3x²(-Cos 2x)/4 + 6x(-Sin 2x)/8 - 6(Cos 2x)/16
=(x³Sin 2x)/2 + 3/4 *(x²Cos 2x) - 6/8 *(xSin 2x) - 6/16 *(Cos 2x) + c

Problem on Factorization

Topic : Factorization

Problem : Solve a^4 +3a^3+27a+81

Solution :
a^4 +3a^3+27a+81
First let's group the terms
(a^4 +3a^3)+(27a+81)
Now we take common factors
from each parenthesis.
a^3(a+3)+ 27(a+3) as 81/27 = 3
Again we have a common factor
(a+3)
So, (a+3)(a^3+27)
or (a+3)(a^3+3^3)
Applying the formula x^3+y^3 = (x+y)(x^2-xy+y^2) for a^3+b^3
we get (a+3)(a+3)(a^2-3a+3^2)
(a+3)(a+3)(a^2-3a+9)
or
(a+3)^2(a^2-3a+9)