Evaluate the Integral integral of 1/(3x-2) with respect to x

Problem

(∫_^)(1/(3*x−2)*d(x))

Solution

1. Identify the form of the integral, which resembles the rule (∫_^)(1/u*d(u))=ln(u)+C

2. Apply a substitution by letting u=3*x−2

3. Differentiate u with respect to x to find d(u)=3*d(x) which implies d(x)=1/3*d(u)

4. Substitute the variables into the integral:

(∫_^)(1/u⋅1/3*d(u))

5. Factor out the constant 1/3 from the integral:

1/3*(∫_^)(1/u*d(u))

6. Integrate with respect to u

1/3*ln(u)+C

7. Back-substitute u=3*x−2 to get the final expression in terms of x

Final Answer

(∫_^)(1/(3*x−2)*d(x))=1/3*ln(3*x−2)+C

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