Evaluate the Integral

Problem

(∫_0^1)((x2+7)*e(−x)*d(x))

Solution

1. Identify the method of integration by parts, which states (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) We will use the tabular method (or repeated integration by parts) for the polynomial (x2+7) and the exponential e(−x)

2. Differentiate the polynomial part until it reaches zero:

   u=x2+7

   u′=2*x

   u″=2

   u‴=0

3. Integrate the exponential part the same number of times:

   d(v)=e(−x)*d(x)

   v=−e(−x)

   (∫_^)(v*d(x))=e(−x)

   (∫_^)(((∫_^)(v*d(x)))*d(x))=−e(−x)

4. Combine the terms using alternating signs to find the antiderivative:

(∫_^)((x2+7)*e(−x)*d(x))=(x2+7)*(−e(−x))−(2*x)*(e(−x))+(2)*(−e(−x))

5. Simplify the antiderivative expression by factoring out −e(−x)

−e(−x)*(x2+7+2*x+2)

−e(−x)*(x2+2*x+9)

6. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 1 and lower limit 0

[−e(−1)*(1+2*(1)+9)]−[−e0*(0+2*(0)+9)]

7. Calculate the numerical values:

−e(12)(−1)−(−1*(9))

−12*e(−1)+9

Final Answer

(∫_0^1)((x2+7)*e(−x)*d(x))=9−12/e

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