Evaluate the Integral integral of 7xe^x with respect to x

Problem

(∫_^)(7*x*ex*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function 7*x and a transcendental function ex use integration by parts.

2. Choose the parts for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=7*x and d(v)=ex*d(x)

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

d(u)/d(x)=7⇒d(u)=7*d(x)

v=(∫_^)(ex*d(x))=ex

4. Apply the integration by parts formula.

(∫_^)(7*x*ex*d(x))=(7*x)*(ex)−(∫_^)(ex*(7*d(x)))

5. Simplify the integral term by moving the constant outside.

(∫_^)(7*x*ex*d(x))=7*x*ex−7*(∫_^)(ex*d(x))

6. Evaluate the remaining integral and add the constant of integration C

(∫_^)(7*x*ex*d(x))=7*x*ex−7*ex+C

7. Factor out the common term 7*ex to simplify the final expression.

(∫_^)(7*x*ex*d(x))=7*ex*(x−1)+C

Final Answer

(∫_^)(7*x*ex*d(x))=7*x*ex−7*ex+C

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