Evaluate the Integral

Problem

(∫_0^1)((x2+5)*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+5) and the exponential e(−x)

2. Differentiate the polynomial part until it reaches zero:

   (u_0)=x2+5

   (u_1)=2*x

   (u_2)=2

   (u_3)=0

3. Integrate the exponential part e(−x) 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+5)*e(−x)*d(x))=(x2+5)*(−e(−x))−(2*x)*(e(−x))+(2)*(−e(−x))

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

−e(−x)*(x2+5+2*x+2)

−e(−x)*(x2+2*x+7)

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)+7)]−[−e0*(0+2*(0)+7)]

7. Calculate the numerical values:

−e(10)(−1)−(−1*(7))

−10*e(−1)+7

Final Answer

(∫_0^1)((x2+5)*e(−x)*d(x))=7−10/e

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