Evaluate the Integral

Problem

(∫_0^1)((x2+2)*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=x2+2 and d(v)=e(−x)*d(x)

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

u=x2+2⇒d(u)=2*x*d(x)

d(v)=e(−x)*d(x)⇒v=−e(−x)

3. Apply the integration by parts formula.

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

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

4. Perform integration by parts again for the remaining integral (∫_^)(x*e(−x)*d(x)) Let u=x and d(v)=e(−x)*d(x)

u=x⇒d(u)=d(x)

d(v)=e(−x)*d(x)⇒v=−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. Substitute the result back into the main expression to find the indefinite integral.

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

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

(∫_^)((x2+2)*e(−x)*d(x))=(−x2−2*x−4)*e(−x)

6. Evaluate the definite integral from 0 to 1 using the Fundamental Theorem of Calculus.

(∫_0^1)((x2+2)*e(−x)*d(x))=[(−x2−2*x−4)*e(−x)]10

=(−(1)2−2*(1)−4)*e(−1)−(−(0)2−2*(0)−4)*e0

=−7*e(−1)−(−4)*(1)

=4−7*e(−1)

Final Answer

(∫_0^1)((x2+2)*e(−x)*d(x))=4−7/e

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