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

Problem

(∫_^)(1/(3−2*x)*d(x))

Solution

1. Identify the form of the integral as a reciprocal function, which suggests a logarithmic result.

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

3. Calculate the differential d(u) by differentiating u with respect to x which gives d(u)=−2*d(x) or 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 to get −1/2*(∫_^)(1/u*d(u))

6. Integrate using the rule (∫_^)(1/u*d(u))=ln(u)+C resulting in −1/2*ln(u)+C

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

Final Answer

(∫_^)(1/(3−2*x)*d(x))=−1/2*ln(3−2*x)+C

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