Evaluate the Integral

Problem

(∫_^)(2*x2*ln(x)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function 2*x2 and a logarithmic function ln(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=ln(x) and d(v)=2*x2*d(x)

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

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

v=(2*x3)/3

4. Apply the integration by parts formula.

(∫_^)(2*x2*ln(x)*d(x))=ln(x)⋅(2*x3)/3−(∫_^)((2*x3)/3⋅1/x*d(x))

5. Simplify the integral term.

(∫_^)(2*x2*ln(x)*d(x))=(2*x3*ln(x))/3−(∫_^)((2*x2)/3*d(x))

6. Evaluate the remaining integral.

(∫_^)((2*x2)/3*d(x))=2/3⋅(x3)/3=(2*x3)/9

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

(∫_^)(2*x2*ln(x)*d(x))=(2*x3*ln(x))/3−(2*x3)/9+C

Final Answer

(∫_^)(2*x2*ln(x)*d(x))=(2*x3*ln(x))/3−(2*x3)/9+C

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