Anti-derivatives & Integration Formulas (initial version) Flashcards






















How can we integrate this problem?

By using the power rule. ln x + C only works when the power of x is equal to 1 (x1)




Write the general formula used in integration by parts
- Here, you want to find two things: u and dv
- After identifying those two elements, you need to find du by differentiating u and you need to find v by integrating dv

Decode the acronym LIATE (for identifying the ‘u’ value)
Logs > Inverse trigs > Algebraic > Trigs > Exponential
- Logs: things like “ln” and “log”
- Inverse trigs: things like “cos-1” or “tan-1“…etc
- Algebraic: things like “x” or “x2” or “x3” or polynomials
- Trigs: things like “sin” or “tan” …etc
- Exponential: things like “ex” and “ax”
What are the two main techniques used in solving integration by parts problems?
1- The LIATE rule (helps with identifying the ‘u’ term, then you should proceed with the original by parts formula)
2- The tabular method


Can I use the tabular method for solving all integration by parts problems?
Yes!
You can use it for all IBP problems
How to know if I chose the correct u and dv terms?
Choose your u and dv based on the LIATE rule (should work perfectly for most problems):
1- If your problem is getting simpler, then your choice was correct
2- If the problem is getting more complicated (especially at the ∫ v du part), then you probably have to go back and switch the values of those two (u and dv) and try again
When to use the u-substitution technique?
When the anti-derivative and the derivative of a given function are available in the same problem
When to use the integration by parts technique?
How many possible scenarios we can expect while solving IBP problems with the tabular method?
3 different cases:
- case 1: when the nth derivative of u is equal to zero (after taking the derivative of u for an x number of times). Example: ∫ x2cosx dx
- case 2: when the nth derivative of u will never be zero, but the function of the integral repeats itself in the table. We have to stop as soon as we see the first match (regardless of the sign and/or any multiplied constants). Example: ∫ ex sinx dx
- case 3: when the nth derivative of u will never be zero, but the nth derivative of u keeps getting smaller and smaller. In this case, we have to stop right after finding the first derivative of 2 and consider the first two rows of the table in our final answer. Example: ∫ x3 lnx dx
What is the purpose of using the u-substitution technique?
The ultimate goal of using this technique is to cancel out all Xs in the problem and be left with Us only (hence the name)