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

Problem

(∫_^)(x*e(−3*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 components u=x and d(v)=e(−3*x)*d(x)

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

4. Integrate d(v) to find v=−1/3*e(−3*x)

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

(∫_^)(x*e(−3*x)*d(x))=x*(−1/3*e(−3*x))−(∫_^)(−1/3*e(−3*x)*d(x))

6. Simplify the expression and the remaining integral:

(∫_^)(x*e(−3*x)*d(x))=−1/3*x*e(−3*x)+1/3*(∫_^)(e(−3*x)*d(x))

7. Evaluate the final integral:

(∫_^)(x*e(−3*x)*d(x))=−1/3*x*e(−3*x)−1/9*e(−3*x)+C

Final Answer

(∫_^)(x*e(−3*x)*d(x))=−1/3*x*e(−3*x)−1/9*e(−3*x)+C

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