Evaluate the Integral integral of 1/(7x+3) with respect to x

Problem

(∫_^)(1/(7*x+3)*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=7*x+3

3. Differentiate the substitution to find d(u)=7*d(x) which implies d(x)=1/7*d(u)

4. Substitute the variables into the integral to get (∫_^)(1/u⋅1/7*d(u))

5. Factor out the constant 1/7 to get 1/7*(∫_^)(1/u*d(u))

6. Integrate with respect to u to obtain 1/7*ln(u)+C

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

Final Answer

(∫_^)(1/(7*x+3)*d(x))=1/7*ln(7*x+3)+C

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