Integration by Parts quiz #1 Flashcards
Integration by Parts quiz #1
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What is the integration by parts formula, and when should it be used?
The integration by parts formula is ∫u dv = u·v − ∫v du. It should be used to integrate the product of two functions when substitution and basic integration rules are ineffective. How do you choose u and dv when applying integration by parts?
Choose u as the function that becomes simpler when differentiated, and dv as the function that is easily integrated. A common strategy is to let u be an algebraic function and dv be an exponential or trigonometric function. How is integration by parts applied to definite integrals?
For definite integrals, apply the bounds to both the u·v term and the ∫v du term after performing integration by parts. What should you do if the resulting integral after applying integration by parts is still too complex to solve directly?
If the resulting integral is still complex, repeat the integration by parts process as needed until the integral becomes solvable. What is the tabular method in integration by parts, and when is it useful?
The tabular method organizes repeated integration by parts into a table of derivatives and integrals, making it efficient for integrals requiring multiple applications of the technique, especially when one function becomes zero after several derivatives. What is the integration by parts formula and in what situation should you use it?
The integration by parts formula is ∫u dv = u·v − ∫v du. It should be used when integrating the product of two functions where substitution and basic integration rules do not work. How do you decide which part of the integrand to choose as u and which as dv in integration by parts?
Choose u as the function that becomes simpler when differentiated, and dv as the function that is easily integrated. Typically, algebraic functions are chosen as u and exponential or trigonometric functions as dv. What should you do if, after applying integration by parts, the resulting integral is still too complex to solve directly?
If the resulting integral is still complex, you should repeat the integration by parts process as many times as needed until the integral becomes solvable. This is common when the integrand requires multiple applications of the technique. How is integration by parts applied to definite integrals with bounds?
For definite integrals, after performing integration by parts, you must apply the bounds to both the u·v term and the ∫v du term. This ensures the correct evaluation of the definite integral. What is the tabular method in integration by parts and when is it especially useful?
The tabular method organizes repeated integration by parts into a table of derivatives and integrals, making the process more efficient. It is especially useful when one function becomes zero after several derivatives, allowing for quick computation of the integral.
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