Evaluate the Integral

Problem

(∫_0^1)(x*ex*d(x))

Solution

1. Identify the integration method as integration by parts, using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign the variables for integration by parts by letting u=x and d(v)=ex*d(x)

3. Differentiate u to find d(u)=d(x) and integrate d(v) to find v=ex

4. Apply the integration by parts formula to the definite integral.

(∫_0^1)(x*ex*d(x))=[x*ex]10−(∫_0^1)(ex*d(x))

5. Evaluate the boundary terms for the first part of the expression.

[x*ex]10=(1)*e1−(0)*e0=e

6. Integrate the remaining term (∫_0^1)(ex*d(x))

(∫_0^1)(ex*d(x))=[ex]10

7. Evaluate the boundaries for the second part.

[ex]10=e1−e0=e−1

8. Subtract the results to find the final value.

e−(e−1)=1

Final Answer

(∫_0^1)(x*ex*d(x))=1

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