Q.131. Integration with proof of formula.

Question 131.
a) If f is a polynomial, verify that ∫ f(x)e^x dx = [ f(x) - f '(x) + f "(x) - f "'(x) + ...] e^x + C

b) Evaluate :  ∫ x^6 e^2x dx.

Answer 131.
Remember the following method to solve any problem of the type
∫ f (x) e^x dx
= ∫ [f (x) + f '(x)] - [f '(x) + f "(x)] + [f "(x) + f "'(x)] - ......] e^x dx
Note that terms f '(x), f "(x), f "'(x), .....are added and subtracted.
This is to be done till successive differentiation ends in a constant which is added and subtracted to complete this step.

Next, break up into separate integrals as under
= ∫ [f (x) + f '(x)]e^x dx - ∫ [f '(x) + f "(x)]e^x dx + ∫ [f "(x) + f '"(x)]e^x dx - ....
then use the formula
∫ [f (x) + f '(x)]e^x dx = f (x) e^x to complete the last step as
= f (x) e^x - f '(x) e^x + f "(x) e^x - ....
= [f (x) - f '(x) + f "(x) - f "'(x) + ....] e^x + c. ... ( 1 )

b)
Let 2x = t
=> 2dx = dt
and x = t/2
=> Integration
= (1/2) ∫ (t/2)^6 e^t dt
= (1/2)^7 ∫ t^6 e^t dt

Using the above formula,
Integral
= (1/2)^7 * [t^6 - 6t^5 + 30t^4 - 120t^3 + 360t^2 - 720t + 720] e^t +c
Putting t = 2x,
= (1/2)^7 * [(2x)^6 - 6(2x)^5 + 30(2x)^4 - 120(2x)^3 + 360(2x)^2 - 720(2x) + 720] e^2x + c
= (1/2)^7 * [64x^6 - 192x^5 + 480x^4 - 960x^3 + 1440x^2 - 1440x + 720] e^2x + c
= (1/8) * [4x^6 - 12x^5 + 30x^4 - 60x^3 + 90x^2 - 90x + 45] e^2x + c.

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