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How do you integrate #int x^3 e^(4x) dx # using integration by parts?

Answer 1

Evelyn Agard

Use integration by parts three times.

Each time choose a power of #x# as #u# and #dv = e^(4x) dx#

Each time you integrate, the power on #x# decreases and the coefficient of the next integral changes.

in the end you'll be able to facotr out a denominator and #e^(4x)# to finish with

#1/128e^(4x)(32x^3-24x^2+12x-3)+C#

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Answer 2

Isaiah Allen

To integrate ( \int x^3 e^{4x} , dx ) using integration by parts:

Choose ( u ) and ( dv ): Let ( u = x^3 ) and ( dv = e^{4x} , dx ).
Compute the differentials: ( du = 3x^2 , dx ) and ( v = \frac{1}{4}e^{4x} ).
Apply the integration by parts formula: [ \int u , dv = uv - \int v , du ]

Substituting the values: [ \int x^3 e^{4x} , dx = x^3 \left( \frac{1}{4}e^{4x} \right) - \int \left( \frac{1}{4}e^{4x} \right) (3x^2 , dx) ]
Simplify and integrate the remaining integral: [ = \frac{1}{4}x^3 e^{4x} - \frac{3}{4} \int x^2 e^{4x} , dx ]

Now, you can use integration by parts again for the second integral.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.