Evaluate the Integral

Problem

(∫_0^1)((x2+1)*e(−x)*d(x))

Solution

1. Identify the method of integration by parts, which states (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Set up the first integration by parts by letting u=x2+1 and d(v)=e(−x)*d(x)

3. Calculate the differentials d(u)=2*x*d(x) and v=−e(−x)

4. Apply the integration by parts formula:

(∫_^)((x2+1)*e(−x)*d(x))=−(x2+1)*e(−x)−(∫_^)(−2*x*e(−x)*d(x))

5. Simplify the expression:

(∫_^)((x2+1)*e(−x)*d(x))=−(x2+1)*e(−x)+2*(∫_^)(x*e(−x)*d(x))

6. Apply integration by parts again for the remaining integral (∫_^)(x*e(−x)*d(x)) by letting u=x and d(v)=e(−x)*d(x) which gives d(u)=d(x) and v=−e(−x)

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

7. Evaluate the inner integral:

(∫_^)(x*e(−x)*d(x))=−x*e(−x)−e(−x)

8. Combine all parts to find the general antiderivative:

(∫_^)((x2+1)*e(−x)*d(x))=−(x2+1)*e(−x)+2*(−x*e(−x)−e(−x))

9. Simplify the antiderivative:

(∫_^)((x2+1)*e(−x)*d(x))=−x2*e(−x)−e(−x)−2*x*e(−x)−2*e(−x)

(∫_^)((x2+1)*e(−x)*d(x))=−e(−x)*(x2+2*x+3)

10. Evaluate the definite integral from 0 to 1

(∫_0^1)((x2+1)*e(−x)*d(x))=[−e(−x)*(x2+2*x+3)]10

11. Substitute the upper and lower limits:

(∫_0^1)((x2+1)*e(−x)*d(x))=(−e(−1)*(1+2*(1)+3))−(−e0*(0+2*(0)+3))

12. Calculate the final numerical value:

(∫_0^1)((x2+1)*e(−x)*d(x))=−6*e(−1)+3

Final Answer

(∫_0^1)((x2+1)*e(−x)*d(x))=3−6/e

Want more problems? Check here!