Integration by Parts with Natural Log

Integration by parts is a powerful technique in calculus that allows us to solve integrals involving products of functions. One common scenario where integration by parts proves useful is when dealing with integrals containing natural logarithms.

Let's explore an example to illustrate this concept:

Example:

Suppose we want to find the integral of ∫x ln(x) dx. Here's how we can solve it using integration by parts.

Step 1: Identifying u and dv

First, we need to choose our 'u' and 'dv' components. A helpful mnemonic to remember is 'LIATE':

Logarithmic functions
Inverse trigonometric functions
Algebraic functions
Trigonometric functions
Exponential functions

We choose the function that comes earlier in LIATE as our 'u'. In this case, ln(x) is a logarithmic function, making it our 'u'. The remaining part, x dx, becomes our 'dv'.

Step 2: Differentiate u and Integrate dv

We differentiate 'u' and integrate 'dv':

u = ln(x) ⇒ du = (1/x) dx
dv = x dx ⇒ v = (x^2)/2

Step 3: Apply the Integration by Parts Formula

The integration by parts formula states:

∫u dv = uv - ∫v du

Now, we substitute the values we found:

∫x ln(x) dx = ln(x) * (x^2)/2 - ∫(x^2)/2 * (1/x) dx

Step 4: Simplify and Integrate

Simplifying the expression:

∫x ln(x) dx = (x^2 * ln(x))/2 - ∫(x/2) dx

Integrating the remaining term:

∫x ln(x) dx = (x^2 * ln(x))/2 - (x^2)/4 + C

where 'C' is the constant of integration.

Key Points to Remember

Integration by parts is not always the most straightforward method, but it's a powerful tool to have in your calculus toolbox.
When choosing 'u' and 'dv', consider the LIATE mnemonic to guide your decision.
Remember to include the constant of integration ('C') after solving the integral.

Understanding integration by parts, particularly in the context of natural logarithms, is essential for mastering calculus and solving various integration problems.

Integration by Parts with Natural Logarithms