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

Problem

(∫_^)(1/(5−3*x)*d(x))

Solution

1. Identify the form of the integral as (∫_^)(1/u*d(u)) which suggests a substitution method.

2. Substitute a new variable u for the denominator by letting u=5−3*x

3. Differentiate u with respect to x to find d(u)=−3*d(x) which implies d(x)=−1/3*d(u)

4. Rewrite the integral in terms of u by substituting the expressions for the denominator and d(x)

(∫_^)(1/u⋅(−1/3)*d(u))

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

−1/3*(∫_^)(1/u*d(u))

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

−1/3*ln(u)+C

7. Back-substitute the original expression 5 - 3xƒ*o*r$ to get the final result.

Final Answer

(∫_^)(1/(5−3*x)*d(x))=−1/3*ln(5−3*x)+C

Want more problems? Check here!