The formula for integration by parts, often referred to as the "UV formula," is a technique used to integrate the product of two functions. The formula is derived from the product rule for differentiation and is given by:

\[
\int u \, dv = u \, v - \int v \, du
\]

Where:
- \( u \) is a differentiable function of \( x \).
- \( dv \) is another differentiable function of \( x \).
- \( du \) is the derivative of \( u \) with respect to \( x \).
- \( v \) is the antiderivative of \( dv \).

### Steps for Using Integration by Parts:
1. **Choose** \( u \) and \( dv \) from the integrand \( \int u \, dv \).
2. **Compute** \( du \) by differentiating \( u \), and find \( v \) by integrating \( dv \).
3. **Substitute** into the formula \( \int u \, dv = u \, v - \int v \, du \).
4. **Simplify** and integrate \( \int v \, du \) if possible.

Would you like more details on how to choose \( u \) and \( dv \), or examples of integration by parts?

### Related Questions:
1. How do you choose the best \( u \) and \( dv \) for integration by parts?
2. What are some common examples where integration by parts is used?
3. Can integration by parts be applied more than once in a problem?
4. How does the LIATE rule help in choosing \( u \) for integration by parts?
5. What is the difference between integration by substitution and integration by parts?

**Tip:** The LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) is a helpful mnemonic for choosing \( u \) in integration by parts. Start with \( u \) being a function that comes earlier in this list!

Discover how to apply the integration by parts formula, also known as the UV formula. Learn the step-by-step process to solve integrals involving the product of two functions with the formula ∫ u dv = u v - ∫ v du. Suitable for advanced high school and undergraduate students studying calculus.

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