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, which uses the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Choose the variables for substitution by letting 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. Simplify the expression and prepare the remaining integral:

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

7. Evaluate the final integral:

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

8. Factor out common terms to simplify the final result:

(∫_^)(x*e(−5*x)*d(x))=−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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