Evaluate the Integral integral of x^2*e^(-x) with respect to x

Problem

(∫_^)(x2*e(−x)*d(x))

Solution

1. Identify the method of integration by parts, which states (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose u=x2 and d(v)=e(−x)*d(x) which implies d(u)=2*x*d(x) and v=−e(−x)

3. Apply the integration by parts formula for the first time.

(∫_^)(x2*e(−x)*d(x))=−x2*e(−x)−(∫_^)(−2*x*e(−x)*d(x))

4. Simplify the expression by pulling out the constant.

(∫_^)(x2*e(−x)*d(x))=−x2*e(−x)+2*(∫_^)(x*e(−x)*d(x))

5. Apply integration by parts again for the remaining integral (∫_^)(x*e(−x)*d(x)) with u=x and d(v)=e(−x)*d(x) so d(u)=d(x) and v=−e(−x)

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

6. Evaluate the inner integral.

(∫_^)(x*e(−x)*d(x))=−x*e(−x)−e(−x)

7. Substitute this result back into the main equation.

(∫_^)(x2*e(−x)*d(x))=−x2*e(−x)+2*(−x*e(−x)−e(−x))+C

8. Distribute the constant and factor out the common term −e(−x)

(∫_^)(x2*e(−x)*d(x))=−x2*e(−x)−2*x*e(−x)−2*e(−x)+C

Final Answer

(∫_^)(x2*e(−x)*d(x))=−e(−x)*(x2+2*x+2)+C

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