Question - Finding the Integral of a Simple Rational Function
Solution:
\(\int f(x) \,dx = \int \frac{1}{3x} \,dx\)
\(= \frac{1}{3} \int \frac{1}{x} \,dx\)
\(= \frac{1}{3} \ln|x| + C\)
where \(C\) is the constant of integration.
Tutor Solution:
To solve the indefinite integral of the function \( f(x) = \frac{1}{3x} \), we will integrate the function with respect to \( x \).
The integral of \( f(x) = \frac{1}{3x} \) can be written as:
\[ \int f(x) \, dx = \int \frac{1}{3x} \, dx \]To simplify the integral, we can factor out constants from the integral. Since \( \frac{1}{3} \) is a constant, we can take it outside the integral:
\[ \int \frac{1}{3x} \, dx = \frac{1}{3} \int \frac{1}{x} \, dx \]The integral \( \int \frac{1}{x} \, dx \) is a well-known integral that results in the natural logarithm of the absolute value of \( x \). Thus, we have:
\[ \frac{1}{3} \int \frac{1}{x} \, dx = \frac{1}{3} \ln |x| \]When we integrate, we must also add an arbitrary constant of integration, which we usually denote by \( C \). Therefore, the final result of the integral becomes:
\[ \int f(x) \, dx = \frac{1}{3} \ln |x| + C \]So, the result of the indefinite integral of the given function \( f(x) = \frac{1}{3x} \) is:
\[ F(x) = \frac{1}{3} \ln |x| + C \]where \( C \) is the constant of integration that represents the family of all antiderivatives of \( f(x) \).
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com