Evaluate the Integral integral of xe^(5x) with respect to x

Problem

(∫_^)(x*e(5*x)*d(x))

Solution

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

2. Choose the components u=x and d(v)=e(5*x)*d(x)

3. Differentiate u to find d(u)=d(x)

4. Integrate d(v) to find v=1/5*e(5*x)

5. Substitute these components into the integration by parts formula:

(∫_^)(x*e(5*x)*d(x))=x*(1/5*e(5*x))−(∫_^)(1/5*e(5*x)*d(x))

6. Evaluate the remaining integral:

(∫_^)(1/5*e(5*x)*d(x))=1/25*e(5*x)

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

1/5*x*e(5*x)−1/25*e(5*x)+C

Final Answer

(∫_^)(x*e(5*x)*d(x))=1/5*x*e(5*x)−1/25*e(5*x)+C

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