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

Problem

(∫_^)(1/(2*x+1)*d(x))

Solution

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

2. Apply a substitution by letting u=2*x+1

3. Differentiate u with respect to x to find d(u)=2*d(x) which implies d(x)=1/2*d(u)

4. Substitute the variables into the integral to get (∫_^)(1/u⋅1/2*d(u))

5. Factor out the constant 1/2 to get 1/2*(∫_^)(1/u*d(u))

6. Integrate with respect to u to obtain 1/2*ln(u)+C

7. Back-substitute u=2*x+1 to express the final answer in terms of x

Final Answer

(∫_^)(1/(2*x+1)*d(x))=1/2*ln(2*x+1)+C

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