Integration by Parts Worksheet/Game

Integration by Parts Worksheet/Game
Welcome to Integration by Parts Worksheet/Game. This game to use the Integration by Parts calculus technique to evaluate integrals of product functions by using the formula \(\int u dv = uv - \int v du \). Derived from the product rule of differentiation, it allows complex integrals to be broken into simpler parts by choosing a function u to differentiate and a function dv to integrate. Scroll down the page for a more detailed explanation.

 

 

How to Play: The Parts Paradigm
The objective of the game is to solve indefinite integrals that cannot be evaluated using basic rules or simple u-substitution. You are applying the Integration by Parts formula:
\(\int u \, dv = uv - \int v \, du\)

The Gameplay Loop

1. Analyze the Product: You will be presented with an integral that is a product of two functions (e.g., x and cos(x)).
   

2. Pick the Solution: From the grid, select the expression that correctly represents the final integrated form.
   

3. Accuracy & Progress: You have 10 “Projects” (problems) to complete. Your accuracy is tracked in the top-right corner.
   

4. Show Proof: If you choose incorrectly, the “Show Proof” button unlocks. Use this to see the step-by-step breakdown of how u and dv were chosen.
   

5. Abort: If a problem is too complex, you can “Abort to Menu” to reset.
   

Understanding Integration by Parts
Integration by Parts is essentially the Product Rule for derivatives, but in reverse. It is used when you are integrating the product of two different types of functions.

1. Choosing u and dv
   The most critical part of the process is deciding which part of your integral is u and which is dv.
   A common mnemonic used is LIATE:
   L - Logarithmic functions (ln x)
   I - Inverse Trigonometric functions (arctan x)
   A - Algebraic functions (x², 3x)
   T - Trigonometric functions (sin x)
   E - Exponential functions (e^x)
   Rule of Thumb: Pick u to be the function that appears higher on the LIATE list. This is because these functions usually become “simpler” when you take their derivative (du).
   

2. The Process
   Once you have assigned u and dv:
   Differentiate u to find du.
   Integrate dv to find v.
   Plug them into the formula: \(uv - \int v \, du\).
   Evaluate the remaining integral: The goal is for \(\int v \, du\) to be much easier to solve than your original problem.
   

Tip for the Game
If you see an algebraic term like x² paired with e^x, you may have to apply the formula twice. The game includes “Repeated Parts” problems where the first pass simplifies the power of x, and the second pass finishes the integration.

Integration by Parts

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