Evaluate the Integral

Problem

(∫_^)(16*x*ln(2*x)*d(x))

Solution

1. Identify the integration method. Since the integrand is a product of an algebraic function 16*x and a logarithmic function ln(2*x) use integration by parts.

2. Choose the parts for integration by parts using the formula (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u)) Let u=ln(2*x) and d(v)=16*x*d(x)

3. Differentiate u to find d(u)

d(u)/d(x)=1/(2*x)⋅2=1/x

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

4. Integrate d(v) to find v

v=(∫_^)(16*x*d(x))=8*x2

5. Apply the integration by parts formula.

(∫_^)(16*x*ln(2*x)*d(x))=(8*x2)*(ln(2*x))−(∫_^)(8*x2⋅1/x*d(x))

6. Simplify the integral term.

(∫_^)(8*x2⋅1/x*d(x))=(∫_^)(8*x*d(x))

7. Evaluate the remaining integral.

(∫_^)(8*x*d(x))=4*x2

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

8*x2*ln(2*x)−4*x2+C

Final Answer

(∫_^)(16*x*ln(2*x)*d(x))=8*x2*ln(2*x)−4*x2+C

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