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

Problem

(∫_^)(1/(2*x−1)*d(x))

Solution

1. Identify the form of the integral, which matches the pattern (∫_^)(1/u*d(u))

2. Apply substitution by letting u=2*x−1

3. Calculate the differential d(u) by differentiating u with respect to x which gives d(u)/d(x)=2

4. Rearrange the differential to solve for d(x) resulting in d(x)=1/2*d(u)

5. Substitute u and d(x) into the original integral to get (∫_^)(1/u⋅1/2*d(u))

6. Factor out the constant 1/2 from the integral.

7. Integrate using the rule (∫_^)(1/u*d(u))=ln(u)+C

8. Back-substitute the original expression for u to find the final result.

Final Answer

(∫_^)(1/(2*x−1)*d(x))=1/2*ln(2*x−1)+C

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