Integration by parts formula (IBP)

📚 What is Integration by Parts?

Integration by parts (IBP) is a technique used to integrate the product of two functions. It's especially helpful when other integration methods, like u-substitution, don't quite do the trick. Essentially, it's the integral version of the product rule for differentiation.

📜 History and Background

The fundamental idea behind integration by parts can be traced back to Brook Taylor in 1715. However, the formula as we know it today was popularized and used extensively by mathematicians like Euler and Bernoulli in the 18th century.

🔑 The Integration by Parts Formula

The integration by parts formula is given by:

$\int u \, dv = uv - \int v \, du$

Where:

• 🔍 $u$ is a function we choose to differentiate.
• 💡 $dv$ is a function we choose to integrate.
• 📝 $du$ is the derivative of $u$.
• 📈 $v$ is the integral of $dv$.

💡 Key Principles and the LIATE mnemonic

The key to successfully using integration by parts lies in choosing appropriate functions for $u$ and $dv$. A helpful mnemonic for this selection is LIATE:

• L: Logarithmic functions (e.g., $\ln(x)$)
• I: Inverse trigonometric functions (e.g., $\arctan(x)$)
• A: Algebraic functions (e.g., $x^2$, $3x+1$)
• T: Trigonometric functions (e.g., $\sin(x)$, $\cos(x)$)
• E: Exponential functions (e.g., $e^x$, $2^x$)

Choose $u$ to be the function that comes earlier in the LIATE list. This generally simplifies the integral. Sometimes you will need to use integration by parts multiple times.

🧮 Real-World Examples

Example 1: $\int x \cos(x) \, dx$

1. 🍎 Let $u = x$ and $dv = \cos(x) \, dx$.
2. 🍌 Then $du = dx$ and $v = \sin(x)$.
3. 🍇 Applying the formula: $\int x \cos(x) \, dx = x\sin(x) - \int \sin(x) \, dx$.
4. 🍉 Integrating the remaining term: $x\sin(x) - (-\cos(x)) + C$.

Therefore, $\int x \cos(x) \, dx = x\sin(x) + \cos(x) + C$.

Example 2: $\int \ln(x) \, dx$

1. 🍎 Let $u = \ln(x)$ and $dv = dx$.
2. 🍌 Then $du = \frac{1}{x} \, dx$ and $v = x$.
3. 🍇 Applying the formula: $\int \ln(x) \, dx = x\ln(x) - \int x \cdot \frac{1}{x} \, dx$.
4. 🍉 Simplifying and integrating: $x\ln(x) - \int dx = x\ln(x) - x + C$.

Therefore, $\int \ln(x) \, dx = x\ln(x) - x + C$.

📝 Conclusion

Integration by parts is a powerful technique for evaluating integrals involving products of functions. By carefully choosing $u$ and $dv$, you can often simplify complex integrals into more manageable forms. Remember the LIATE mnemonic to help guide your selection of $u$. Practice makes perfect! 🚀