Integration by Parts
Integration by Parts is one of the most powerful techniques in calculus for evaluating integrals of products of functions. It essentially "undoes" the Product Rule for differentiation.
🔑 The Origin: From the Product Rule
Recall the Product Rule for differentiation:
$$ \frac{d}{dx}[u \cdot v] = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx} $$
Now, integrate both sides with respect to $x$:
$$ u \cdot v = \int u \, dv + \int v \, du $$
Rearranging, we get the Integration by Parts formula:
$$ \boxed{\int u \, dv = u \cdot v - \int v \, du} $$
This formula transforms a difficult integral $\int u \, dv$ into a (hopefully) simpler integral $\int v \, du$.
📋 The LIATE Rule
The hardest part is choosing what to call $u$ and what to call $dv$. The LIATE mnemonic helps you pick $u$:
| Letter | Function Type | Example |
|---|---|---|
| L | Logarithmic | $\ln x, \log x$ |
| I | Inverse Trig | $\arctan x, \arcsin x$ |
| A | Algebraic | $x^2, 3x, x^n$ |
| T | Trigonometric | $\sin x, \cos x$ |
| E | Exponential | $e^x, 2^x$ |
Rule: Choose $u$ as the function that appears earliest in LIATE. The remaining factor becomes $dv$.
⚙️ Step-by-Step Process
- Identify the integral as a product of two different types of functions.
- Choose $u$ using LIATE — pick the function that simplifies when differentiated.
- Set $dv$ as everything else (including $dx$).
- Compute $du$ by differentiating $u$.
- Compute $v$ by integrating $dv$.
- Substitute into the formula: $\int u\,dv = uv - \int v\,du$
- Evaluate the remaining integral $\int v\,du$.