Q.79. Integration technique

Question 79.
Find ∫ t^3 e^t dt

Answer 79.
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 )

Now, observe how the above problem is tackled.
∫ t^3 e^t dt
= ∫ [(t^3 + 3t^2) - (3t^2 + 6t) + (6t + 6) - 6] e^t dt
= ∫ (t^3 + 3t^2)e^t dt - ∫ (3t^2 + 6t)e^t dt + ∫ (6t + 6)e^t dt - 6 ∫ e^t dt
= t^3 e^t - 3t^2 e^t + 6t e^t - 6 e^t + c.
= (t^3 - 3t^2 + 6t - 6) e^t + c.

Using ( 1 ) as a formula, one can orally write down the answer also.

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