Evaluate the Integral integral of x^3e^x with respect to x

Problem

(∫_^)(x3*ex*d(x))

Solution

1. Identify the method of Integration by Parts, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign variables for the first application: let u=x3 and d(v)=ex*d(x)

3. Calculate the differentials: d(u)=3*x2*d(x) and v=ex

4. Apply the formula:

(∫_^)(x3*ex*d(x))=x3*ex−(∫_^)(3*x2*ex*d(x))

5. Repeat Integration by Parts for (∫_^)(3*x2*ex*d(x)) let u=3*x2 and d(v)=ex*d(x) so d(u)=6*x*d(x) and v=ex

(∫_^)(x3*ex*d(x))=x3*ex−(3*x2*ex−(∫_^)(6*x*ex*d(x)))

6. Repeat Integration by Parts for (∫_^)(6*x*ex*d(x)) let u=6*x and d(v)=ex*d(x) so d(u)=6*d(x) and v=ex

(∫_^)(x3*ex*d(x))=x3*ex−3*x2*ex+(6*x*ex−(∫_^)(6*ex*d(x)))

7. Evaluate the final integral: (∫_^)(6*ex*d(x))=6*ex

(∫_^)(x3*ex*d(x))=x3*ex−3*x2*ex+6*x*ex−6*ex+C

8. Factor out the common term ex to simplify the expression.

Final Answer

(∫_^)(x3*ex*d(x))=ex*(x3−3*x2+6*x−6)+C

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