Evaluate the Integral integral of xe^(8x) with respect to x

Problem

(∫_^)(x*e(8*x)*d(x))

Solution

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

2. Choose the variables for substitution by letting u=x and d(v)=e(8*x)*d(x)

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

4. Integrate d(v) to find v=(∫_^)(e(8*x)*d(x))=1/8*e(8*x)

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

(∫_^)(x*e(8*x)*d(x))=x*(1/8*e(8*x))−(∫_^)(1/8*e(8*x)*d(x))

6. Evaluate the remaining integral:

(∫_^)(1/8*e(8*x)*d(x))=1/8⋅1/8*e(8*x)=1/64*e(8*x)

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

1/8*x*e(8*x)−1/64*e(8*x)+C

Final Answer

(∫_^)(x*e(8*x)*d(x))=1/8*x*e(8*x)−1/64*e(8*x)+C

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