Integration by parts is a technique used to integrate the product of two functions. It's essentially the reverse of the product rule for differentiation. The formula is given by: $\int u \, dv = uv - \int v \, du$. The key is choosing appropriate $u$ and $dv$ to simplify the integral. A helpful mnemonic is LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to guide your choice of $u$.
When applying integration by parts, you're aiming to transform a complex integral into something more manageable. Sometimes, you might need to apply integration by parts multiple times to fully solve the integral. Remember to include the constant of integration, $+C$, for indefinite integrals!
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Integration by Parts | A. A function chosen for differentiation in integration by parts. |
| 2. $u$ | B. The function remaining after choosing $u$ in integration by parts, including $dx$. |
| 3. $dv$ | C. A technique to integrate the product of two functions. |
| 4. $v$ | D. The integral of $dv$. |
| 5. LIATE | E. Acronym to help choose 'u' in integration by parts: Logarithmic, Inverse Trig, Algebraic, Trig, Exponential. |
Integration by parts is used to integrate the ________ of two functions. The formula is $\int u \, dv = uv - \int v \, du$. The goal is to choose $u$ and $dv$ such that the new integral, $\int v \, du$, is ________ than the original. The mnemonic ________ can help in selecting $u$. Remember to add ________ after evaluating indefinite integrals.
Explain a situation where you would need to apply integration by parts more than once to solve an integral. Give an example function and explain why it requires multiple applications.
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