Evaluate the Integral integral of xe^(6x) with respect to x

Problem

(∫_^)(x*e(6*x)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function x and an exponential function e(6*x) use integration by parts.

2. Choose the variables for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=x and d(v)=e(6*x)*d(x)

3. Differentiate u and integrate d(v) to find d(u) and v

d(u)=d(x)

v=(∫_^)(e(6*x)*d(x))=1/6*e(6*x)

4. Apply the integration by parts formula.

(∫_^)(x*e(6*x)*d(x))=x*(1/6*e(6*x))−(∫_^)(1/6*e(6*x)*d(x))

5. Evaluate the remaining integral.

(∫_^)(1/6*e(6*x)*d(x))=1/6⋅1/6*e(6*x)=1/36*e(6*x)

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

1/6*x*e(6*x)−1/36*e(6*x)+C

Final Answer

(∫_^)(x*e(6*x)*d(x))=1/6*x*e(6*x)−1/36*e(6*x)+C

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