Evaluate the Integral integral of e^xx with respect to x

Problem

(∫_^)(ex*x*d(x))

Solution

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

2. Apply the formula for integration by parts, which is (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

3. Choose the variables u and d(v) using the LIATE rule. Let u=x and d(v)=ex*d(x)

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

d(u)=d(x)

v=(∫_^)(ex*d(x))=ex

5. Substitute these into the integration by parts formula.

(∫_^)(ex*x*d(x))=x*ex−(∫_^)(ex*d(x))

6. Evaluate the remaining integral.

(∫_^)(ex*d(x))=ex

7. Combine the terms and add the constant of integration C

x*ex−ex+C

8. Factor out the common term ex to simplify the expression.

ex*(x−1)+C

Final Answer

(∫_^)(ex*x*d(x))=ex*(x−1)+C

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