Evaluate the Integral

Problem

(∫_0^1)((x2+8)*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+8) and the exponential e(−x)

2. Differentiate the polynomial part until it reaches zero:

(u_0)=x2+8

(u_1)=2*x

(u_2)=2

(u_3)=0

3. Integrate the exponential part repeatedly:

d(v)=e(−x)*d(x)

(v_1)=−e(−x)

(v_2)=e(−x)

(v_3)=−e(−x)

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

(∫_^)((x2+8)*e(−x)*d(x))=(x2+8)*(−e(−x))−(2*x)*(e(−x))+(2)*(−e(−x))

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

−e(−x)*(x2+8+2*x+2)

−e(−x)*(x2+2*x+10)

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)+10)]−[−e0*(0+2*(0)+10)]

7. Calculate the numerical values:

−e(13)(−1)−(−1)*(10)

−13*e(−1)+10

Final Answer

(∫_0^1)((x2+8)*e(−x)*d(x))=10−13/e

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