How many times must you apply integration by parts in order to find the antiderivative of x2 e4x ?

Question

How many times must you apply integration by parts in order to find the antiderivative of x² e^4x ?

A	2
B	4
C	More than 4
D	Integration by parts is not necessary

Answer 1

If you just need to know how many times to use Integration by parts, you can use the DI method and count how many times you have ∫f(x)g(x)dx.

F(x) = ∫x²e^4xdx

Let's differentiate x² and integrate e^4x

D I

1) + x² e^4x

2) - 2x e^4x/4

3) + 2 e^4x/16

4) - 0 e^4x/64

By the rules of the DI method, line 3), going left to right, you have 2∫e^4xdx/16

So there were only 2 repetitions before a simple integration.

To get the answer just accumulate the diagonals and add C.

F(x) = x²e^4x/4 - 2xe^4x/16 + 2e^4x/64 + C

= x²e^4x/4 - xe^4x/8 + e^4x/32 + C

Answer 2