Question

Use Exercise 36 to find \(\int {{x^4}{e^x}dx} \)

Short answer

The value of \(\int {{x^4}{e^x}dx} \) is \({e^x}({x^4} - 4{x^3} + 12{x^2} - 24x + 24) + c\)

Step-by-step solution

Step-1:Given Data

Given data is \(\int {{x^4}{e^x}dx} \)

Step-2: Evaluation

Evaluating integration by using parts:

\(\int {udv = uv + \int {vdu \to (1)} } \)

Let \(u = {x^4}\) and \(dv = {e^x}dx\)

Differentiating with respect to ‘x’

Step-3: Substitution

Substituting the above values in Equation (1)

Step-4: Putting in the values

Finding the values of \(\int {{e^x}{x^3}dx} \) by using integration parts:

\(\int {udv = uv - \int {vdu} } \)

Let \(u = {x^3}\)

Differentiating with respect to ‘x’

Step-5: Finding the values of integration by using integration parts:

\(\int {udv = uv - \int {vdu} } \)

Let and

Step-6: Again finding \(\int {x{e^x}dx} \) by using integration by parts:

To find:- \(\int {x{e^x}dx} \)

Let \(u = x\)

Differentiating with respect to ‘x’

Substituting the values in below-given equation

Substituting the value in Equation (6)

Substituting equation (7) in equation (5)

Step-7: Substituting equation (8) in equation (4)

Hence the answer for given integration is

\(\int {{x^4}{e^x}dx = {e^x}({x^4} - 4{x^3} + 12{x^2} - 24x + 24) + c} \)