Evaluate the Integral

Problem

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

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

u=x^2+2⇒d(u)=2*x*d(x)

d(v)=e^(−x)*d(x)⇒v=−e^(−x)

3. Apply the integration by parts formula.

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

(∫_^)((x^2+2)*e^(−x)*d(x))=−(x^2+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.

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

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

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

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

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

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

=−7*e^(−1)−(−4)*(1)

=4−7*e^(−1)

Final Answer

(∫_0^1)((x^2+2)*e^(−x)*d(x))=4−7/e

Want more problems? Check here!