Evaluate the Integral integral of 1/(6x+4) with respect to x

Problem

(∫_^)(1/(6*x+4)*d(x))

Solution

1. Identify the form of the integral as a reciprocal of a linear function, which suggests using a usubstitution.

2. Substitute u=6*x+4 to simplify the denominator.

3. Differentiate the substitution to find d(u)=6*d(x) which implies d(x)=1/6*d(u)

4. Rewrite the integral in terms of u

(∫_^)(1/u⋅1/6*d(u))

5. Factor out the constant 1/6 from the integral:

1/6*(∫_^)(1/u*d(u))

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

1/6*ln(u)+C

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

1/6*ln(6*x+4)+C

Final Answer

(∫_^)(1/(6*x+4)*d(x))=1/6*ln(6*x+4)+C

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