Integrate 1/3x.
Understand the Problem
The question is asking to find the integral of the function 1/(3x). This is a calculus problem where we will apply the rules of integration to compute the antiderivative of the given expression.
Answer
The integral is \( \int \frac{1}{3x} \, dx = \frac{1}{3} \ln |x| + C \).
Answer for screen readers
The integral of ( \frac{1}{3x} ) is: $$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln |x| + C $$
Steps to Solve
- Identify the Function to Integrate
We need to find the integral of the function ( \frac{1}{3x} ).
- Simplify the Expression
The function can be rewritten using a constant factor: $$ \int \frac{1}{3x} , dx = \frac{1}{3} \int \frac{1}{x} , dx $$
- Apply the Integration Rule
Recall that the integral of ( \frac{1}{x} ) is ( \ln |x| + C ), where ( C ) is the constant of integration. Thus, we have: $$ \frac{1}{3} \int \frac{1}{x} , dx = \frac{1}{3} (\ln |x| + C) $$
- Combine the Results
Distributing the constant ( \frac{1}{3} ) gives us: $$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln |x| + C $$
The integral of ( \frac{1}{3x} ) is: $$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln |x| + C $$
More Information
When integrating functions, it's essential to remember to include the constant of integration ( C ) because integrals represent a family of functions differing by a constant. The logarithmic function arises frequently in integrals, especially involving ( \frac{1}{x} ).
Tips
- Forgetting to write the constant of integration ( C ).
- Confusing the rules of integration; particularly, remembering that ( \int \frac{1}{x} , dx ) leads to a logarithmic form.
