Integration by Parts, Formula, Rule, Example, Order - Calculus

Integration by parts is a calculus technique to integrate the product of two functions, derived from the product rule for differentiation. It transforms a complex integral into simpler ones, using the formula: ∫u dv = uv - ∫v du. To apply it, you select a function to be 'u' (which will be differentiated) and the rest as 'dv' (which will be integrated), often guided by the LIATE mnemonic to choose 'u' such that its derivative simplifies, and 'dv' such that its integral is easily found.

💡The Formula
The integration by parts formula is: ∫u dv = uv - ∫v du

💡When to Use It
This method is particularly useful for integrating products of functions, such as:
• A logarithmic function multiplied by an algebraic function.
• A trigonometric function multiplied by an algebraic function.
• An algebraic function multiplied by an exponential function.

💡Steps to Apply the Formula
• Identify u and dv: From the product of functions to be integrated, choose a function 'u' and the remaining part as 'dv'.
• Find du and v: Differentiate 'u' to find 'du' and integrate 'dv' to find 'v'.
• Apply the formula: Substitute 'u', 'v', 'du', and 'dv' into the integration by parts formula.
• Simplify: Evaluate the new, hopefully simpler, integral.

💡How to Choose u and dv (The LIATE Rule)
The acronym LIATE can help decide which function to choose for 'u':
• L: ogarithmic functions (e.g., ln(x))
• I: nverse trigonometric functions (e.g., arcsin(x))
• A: lgebraic functions (e.g., x, x²)
• T: rigonometric functions (e.g., sin(x), cos(x))
• E: xponential functions (e.g., e^x)
The function that appears earliest in this list should generally be chosen as 'u'.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1nyZAxMFIv3phTs8a9uQiZ7WKvrQ4fBKt/view?usp=drive_link
• Answers: https://drive.google.com/file/d/16wbbNzLH-O0LGu6SEvM5Nq9gNQlpoR-H/view?usp=drive_link

💡Chapters:
00:00 Integration by parts
01:05 Worked example

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