Evaluate the Integral

Problem

(∫_0^1)((x^2+4)*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 this twice or use the tabular method. Let u=x^2+4 and d(v)=e^(−x)*d(x)

2. Differentiate u and integrate d(v) for the first application:

u=x^2+4⇒d(u)=2*x*d(x)

d(v)=e^(−x)*d(x)⇒v=−e^(−x)

3. Apply the integration by parts formula:

(∫_^)((x^2+4)*e^(−x)*d(x))=−(x^2+4)*e^(−x)−(∫_^)(−2*x*e^(−x)*d(x))

(∫_^)((x^2+4)*e^(−x)*d(x))=−(x^2+4)*e^(−x)+2*(∫_^)(x*e^(−x)*d(x))

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

(u_2)=x⇒d((u_2))=d(x)

d((v_2))=e^(−x)*d(x)⇒(v_2)=−e^(−x)

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

(∫_^)(x*e^(−x)*d(x))=−x*e^(−x)−e^(−x)

5. Combine the results to find the general antiderivative:

(∫_^)((x^2+4)*e^(−x)*d(x))=−(x^2+4)*e^(−x)+2*(−x*e^(−x)−e^(−x))

(∫_^)((x^2+4)*e^(−x)*d(x))=−x^2*e^(−x)−4*e^(−x)−2*x*e^(−x)−2*e^(−x)

(∫_^)((x^2+4)*e^(−x)*d(x))=(−x^2−2*x−6)*e^(−x)

6. Evaluate the definite integral from 0 to 1

(∫_0^1)((x^2+4)*e^(−x)*d(x))=[(−x^2−2*x−6)*e^(−x)]^1_0

=(−(1)^2−2*(1)−6)*e^(−1)−(−(0)^2−2*(0)−6)*e^0

=−9*e^(−1)−(−6)

=6−9/e

Final Answer

(∫_0^1)((x^2+4)*e^(−x)*d(x))=6−9/e

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