Evaluate the Integral integral of 1/(5x+4) with respect to x

Problem

(∫_^)(1/(5*x+4)*d(x))

Solution

1. Identify the form of the integral, which is a reciprocal of a linear function 1/(a*x+b)

2. Apply the substitution method by letting u=5*x+4

3. Calculate the differential d(u) by differentiating u with respect to x

d(u)=5*d(x)

4. Solve for dx to substitute it back into the integral.

d(x)=1/5*d(u)

5. Substitute u and d(x) into the original integral.

(∫_^)(1/u⋅1/5*d(u))

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

1/5*(∫_^)(1/u*d(u))

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

1/5*ln(u)+C

8. Back-substitute the original expression for u which is 5*x+4

1/5*ln(5*x+4)+C

Final Answer

(∫_^)(1/(5*x+4)*d(x))=1/5*ln(5*x+4)+C

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