Evaluate the Integral

Problem

(∫_0^1)((x2+3)*e(−x)*d(x))

Solution

1. Identify the method of integration by parts, which states (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=x2+3 and d(v)=e(−x)*d(x)

2. Calculate the differentials and integrals for the first application: d(u)=2*x*d(x) and v=−e(−x)

3. Apply the integration by parts formula:

(∫_^)((x2+3)*e(−x)*d(x))=−(x2+3)*e(−x)−(∫_^)(−2*x*e(−x)*d(x))

4. Simplify the expression:

(∫_^)((x2+3)*e(−x)*d(x))=−(x2+3)*e(−x)+2*(∫_^)(x*e(−x)*d(x))

5. Apply integration by parts again for the remaining integral (∫_^)(x*e(−x)*d(x)) Let 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 expression:

(∫_^)((x2+3)*e(−x)*d(x))=−(x2+3)*e(−x)+2*(−x*e(−x)−e(−x))

8. Combine and simplify the indefinite integral:

(∫_^)((x2+3)*e(−x)*d(x))=−e(−x)*(x2+2*x+5)

9. Evaluate the definite integral from 0 to 1 by plugging in the bounds:

[−e(−x)*(x2+2*x+5)]10

10. Calculate the value at the upper bound x=1

−e(−1)*(1+2*(1)+5)=−8*e(−1)

11. Calculate the value at the lower bound x=0

−e0*(0+2*(0)+5)=−5

12. Subtract the lower bound value from the upper bound value:

−8*e(−1)−(−5)=5−8/e

Final Answer

(∫_0^1)((x2+3)*e(−x)*d(x))=5−8/e

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