Techniques of Integration

Integration by Parts

Integration by parts is a fundamental technique used to evaluate integrals where the standard methods such as substitution are not effective. It is especially useful when integrating the product of two functions or functions like the natural logarithm, which do not have straightforward antiderivatives.

• Formula: The integration by parts formula is derived from the product rule for differentiation and is given by:

• Choosing u and dv: Select u as the function that becomes simpler when differentiated, and dv as the function that is easy to integrate.

• Memory Tool (LIATE): A common mnemonic for choosing u is LIATE, which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. The function that appears first in this list should be chosen as u.

Example: Integrating the Natural Logarithm Function

Let us evaluate the integral of the natural logarithm function:

• Step 1: Assign u and dv

  • Let (since its derivative simplifies the expression).

  • Let (since it is easy to integrate).

• Step 2: Compute du and v

• Step 3: Substitute into the Integration by Parts Formula

• Final Answer:

Key Points and Applications

• Integration by parts is essential for integrating products of functions and functions like that do not have elementary antiderivatives.

• Choosing u and dv wisely is crucial for simplifying the integral.

• The LIATE rule helps prioritize which function to select as u.

• This technique is widely used in business calculus for solving integrals involving logarithmic, exponential, and algebraic functions.

Summary Table: LIATE Rule for Choosing u

Additional info: The LIATE rule is a helpful guideline but not a strict rule; always consider which choice simplifies the integral most effectively.