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

Problem

(∫_^)(5*x*e(3*x)*d(x))

Solution

1. Identify the method of integration by parts, which is defined by the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

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

3. Differentiate u to find d(u)=5*d(x) and integrate d(v) to find v=1/3*e(3*x)

4. Apply the formula by substituting the values of u v d(u) and d(v) into the integration by parts equation.

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

5. Simplify the expression and the remaining integral.

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

6. Evaluate the final integral (∫_^)(e(3*x)*d(x))=1/3*e(3*x)

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

7. Combine the terms and factor out common constants if necessary.

(∫_^)(5*x*e(3*x)*d(x))=5/3*x*e(3*x)−5/9*e(3*x)+C

Final Answer

(∫_^)(5*x*e(3*x)*d(x))=5/3*x*e(3*x)−5/9*e(3*x)+C

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