Evaluate the Integral integral of 3xe^(5x) with respect to x

Problem

(∫_^)(3*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. Assign the variables for integration by parts by letting u=3*x and d(v)=e(5*x)*d(x)

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

4. Integrate d(v) to find v=1/5*e(5*x)

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

(∫_^)(3*x*e(5*x)*d(x))=(3*x)*(1/5*e(5*x))−(∫_^)((1/5*e(5*x))*(3)*d(x))

6. Simplify the expression and the remaining integral:

(∫_^)(3*x*e(5*x)*d(x))=3/5*x*e(5*x)−3/5*(∫_^)(e(5*x)*d(x))

7. Evaluate the final integral:

(∫_^)(3*x*e(5*x)*d(x))=3/5*x*e(5*x)−3/5*(1/5*e(5*x))+C

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

(∫_^)(3*x*e(5*x)*d(x))=3/5*x*e(5*x)−3/25*e(5*x)+C

Final Answer

(∫_^)(3*x*e(5*x)*d(x))=3/5*x*e(5*x)−3/25*e(5*x)+C

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