Evaluate the Integral

Problem

(∫_0^ln(2))(x*e^x*d(x))

Solution

1. Identify the method of integration. Since the integrand is a product of an algebraic function x and a transcendental function e^x use integration by parts.

2. Set up the variables for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=x and d(v)=e^x*d(x)

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

d(u)=d(x)

v=e^x

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

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

5. Evaluate the remaining integral of e^x

(∫_^)(x*e^x*d(x))=x*e^x−e^x

6. Apply the fundamental theorem of calculus by evaluating the expression from 0 to ln(2)

(∫_0^ln(2))(x*e^x*d(x))=[x*e^x−e^x]^ln(2)_0

7. Substitute the upper limit ln(2) into the expression.

ln(2)*e^ln(2)−e^ln(2)=ln(2)*(2)−2

=2*ln(2)−2

8. Substitute the lower limit 0 into the expression.

(0)*e^0−e^0=0−1

=−1

9. Subtract the lower limit result from the upper limit result.

(2*ln(2)−2)−(−1)=2*ln(2)−2+1

=2*ln(2)−1

Final Answer

(∫_0^ln(2))(x*e^x*d(x))=2*ln(2)−1

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