Evaluate the Integral integral of xe^(-4x) with respect to x

Problem

(∫_^)(x*e(−4*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 u=x and d(v)=e(−4*x)*d(x)

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

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

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

(∫_^)(x*e(−4*x)*d(x))=x*(−1/4*e(−4*x))−(∫_^)(−1/4*e(−4*x)*d(x))

6. Simplify the expression:

(∫_^)(x*e(−4*x)*d(x))=−x/4*e(−4*x)+1/4*(∫_^)(e(−4*x)*d(x))

7. Evaluate the remaining integral:

(∫_^)(x*e(−4*x)*d(x))=−x/4*e(−4*x)+1/4*(−1/4*e(−4*x))+C

8. Factor out common terms to reach the final form:

(∫_^)(x*e(−4*x)*d(x))=−x/4*e(−4*x)−1/16*e(−4*x)+C

Final Answer

(∫_^)(x*e(−4*x)*d(x))=−1/4*x*e(−4*x)−1/16*e(−4*x)+C

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