Evaluate the Integral

Problem

(∫_^)(ln(5*x−4)*d(x))

Solution

1. Identify the integration method. Since the integrand is a logarithmic function, use integration by parts where (∫_^)(u*d(v))=u*v−(∫_^)(v*d(u))

2. Assign variables for integration by parts. Let u=ln(5*x−4) and d(v)=d(x)

3. Differentiate u and integrate d(v)

d(u)/d(x)=5/(5*x−4)

d(u)=5/(5*x−4)*d(x)

v=x

4. Apply the integration by parts formula.

(∫_^)(ln(5*x−4)*d(x))=x*ln(5*x−4)−(∫_^)((5*x)/(5*x−4)*d(x))

5. Simplify the remaining integral using algebraic manipulation. Rewrite the numerator 5*x as (5*x−4)+4

(∫_^)((5*x)/(5*x−4)*d(x))=(∫_^)((5*x−4+4)/(5*x−4)*d(x))

(∫_^)((5*x−4+4)/(5*x−4)*d(x))=(∫_^)((1+4/(5*x−4))*d(x))

6. Integrate the simplified terms.

(∫_^)(1*d(x))=x

(∫_^)(4/(5*x−4)*d(x))=4/5*ln(5*x−4)

7. Combine all parts and add the constant of integration C

(∫_^)(ln(5*x−4)*d(x))=x*ln(5*x−4)−(x+4/5*ln(5*x−4))+C

(∫_^)(ln(5*x−4)*d(x))=x*ln(5*x−4)−4/5*ln(5*x−4)−x+C

8. Factor the logarithmic terms to simplify the final expression.

(∫_^)(ln(5*x−4)*d(x))=(x−4/5)*ln(5*x−4)−x+C

Final Answer

(∫_^)(ln(5*x−4)*d(x))=(x−4/5)*ln(5*x−4)−x+C

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