Evaluate the Integral integral of xe^(x/3) with respect to x

Problem

(∫_^)(x*e(x/3)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function x and an exponential function e(x/3) use integration by parts.

2. Choose the parts for the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=x and d(v)=e(x/3)*d(x)

3. Differentiate u to find d(u)

d(u)=d(x)

4. Integrate d(v) to find v

v=(∫_^)(e(x/3)*d(x))

v=3*e(x/3)

5. Apply the integration by parts formula.

(∫_^)(x*e(x/3)*d(x))=(x)*(3*e(x/3))−(∫_^)(3*e(x/3)*d(x))

6. Evaluate the remaining integral.

(∫_^)(3*e(x/3)*d(x))=3*(3*e(x/3))

(∫_^)(3*e(x/3)*d(x))=9*e(x/3)

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

(∫_^)(x*e(x/3)*d(x))=3*x*e(x/3)−9*e(x/3)+C

8. Factor the expression to simplify.

(∫_^)(x*e(x/3)*d(x))=3*e(x/3)*(x−3)+C

Final Answer

(∫_^)(x*e(x/3)*d(x))=3*x*e(x/3)−9*e(x/3)+C

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