Integration of log x

Integration is a fundamental concept in calculus that deals with finding the antiderivative of a given function. When it comes to integrating logarithmic functions, we need to use specific integration techniques based on the properties of logarithms.

Logarithmic properties

Before diving into the integration of log x, let's briefly review some important properties of logarithms:

1. Product rule: log(xy) = log(x) + log(y)
2. Quotient rule: log(x/y) = log(x) - log(y)
3. Power rule: log(x^n) = n*log(x)
4. Change of base formula: log_a(x) = log_b(x)/log_b(a)

These properties will come in handy when integrating logarithmic functions.

Integration of log x

The integral of log x, denoted as ∫log(x) dx, can be evaluated using a technique called integration by parts. This technique involves splitting the original function into two parts and applying a specific formula.

Integration by parts formula: ∫u dv = uv - ∫v du

Let's apply this formula to the integral of log x:

We choose:
u = log(x) (to differentiate)
dv = dx (to integrate)

Differentiating u:
du/dx = 1/x

Integrating dv:
v = x

Applying the integration by parts formula:
∫log(x) dx = x*log(x) - ∫x*(1/x) dx
= x*log(x) - ∫dx
= x*log(x) - x + C

Therefore, the integral of log x is x*log(x) - x + C, where C is the constant of integration.

Conclusion

The integration of log x can be evaluated using the integration by parts technique. By applying the formula and simplifying the resulting expression, we find that the integral of log x is x*log(x) - x + C. Understanding the properties of logarithms and the techniques of integration allows us to solve more complex problems involving logarithmic functions.