Evaluate the Integral

Problem

(∫_^)(7*x2*ln(x)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function 7*x2 and a logarithmic function ln(x) use integration by parts.

2. Assign variables for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=ln(x) and d(v)=7*x2*d(x)

3. Differentiate u to find d(u)

d(u)/d(x)=1/x

d(u)=1/x*d(x)

4. Integrate d(v) to find v

v=(∫_^)(7*x2*d(x))

v=(7*x3)/3

5. Apply the integration by parts formula.

(∫_^)(7*x2*ln(x)*d(x))=ln(x)⋅(7*x3)/3−(∫_^)((7*x3)/3⋅1/x*d(x))

6. Simplify the integral term.

(∫_^)(7*x2*ln(x)*d(x))=(7*x3*ln(x))/3−(∫_^)((7*x2)/3*d(x))

7. Evaluate the remaining integral.

(∫_^)((7*x2)/3*d(x))=7/3⋅(x3)/3

(∫_^)((7*x2)/3*d(x))=(7*x3)/9

8. Combine the results and add the constant of integration C

(7*x3*ln(x))/3−(7*x3)/9+C

Final Answer

(∫_^)(7*x2*ln(x)*d(x))=(7*x3*ln(x))/3−(7*x3)/9+C

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