Evaluate the Integral integral of 1/(3x+5) with respect to x

Problem

(∫_^)(1/(3*x+5)*d(x))

Solution

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

2. Apply substitution by letting u=3*x+5

3. Calculate the differential d(u) by differentiating u with respect to x which gives d(u)=3*d(x) or d(x)=1/3*d(u)

4. Substitute the expressions for u and d(x) into the integral to get (∫_^)(1/u⋅1/3*d(u))

5. Factor out the constant 1/3 from the integral.

1/3*(∫_^)(1/u*d(u))

6. Integrate with respect to u using the natural logarithm rule.

1/3*ln(u)+C

7. Back-substitute the original expression 3*x+5 for u to find the final result.

Final Answer

(∫_^)(1/(3*x+5)*d(x))=1/3*ln(3*x+5)+C

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