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

Problem

(∫_^)(x^3*e^x*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=x^3 and d(v)=e^x*d(x)

3. Calculate the differentials: d(u)=3*x^2*d(x) and v=e^x

4. Apply the formula:

(∫_^)(x^3*e^x*d(x))=x^3*e^x−(∫_^)(3*x^2*e^x*d(x))

5. Repeat Integration by Parts for (∫_^)(3*x^2*e^x*d(x)) let u=3*x^2 and d(v)=e^x*d(x) so d(u)=6*x*d(x) and v=e^x

(∫_^)(x^3*e^x*d(x))=x^3*e^x−(3*x^2*e^x−(∫_^)(6*x*e^x*d(x)))

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

(∫_^)(x^3*e^x*d(x))=x^3*e^x−3*x^2*e^x+(6*x*e^x−(∫_^)(6*e^x*d(x)))

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

(∫_^)(x^3*e^x*d(x))=x^3*e^x−3*x^2*e^x+6*x*e^x−6*e^x+C

8. Factor out the common term e^x to simplify the expression.

Final Answer

(∫_^)(x^3*e^x*d(x))=e^x*(x^3−3*x^2+6*x−6)+C

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