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 for the substitution by letting 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=(∫_^)(e(3*x)*d(x))=1/3*e(3*x)

5. Apply the formula for integration by parts.

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

6. Evaluate the remaining integral.

(∫_^)(1/3*e(3*x)*d(x))=1/9*e(3*x)

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

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

Want more problems? Check here!