Integration by parts

Definition

Integration by parts is a technique used to integrate the product of two functions. It is based on the product rule for differentiation and is expressed as $$ \int u \, dv = uv - \int v \, du $$.

5 Must Know Facts For Your Next Test

1. The formula for integration by parts is derived from the product rule of differentiation.
2. Choosing which function to set as \(u\) and which as \(dv\) can significantly impact the ease of solving the integral.
3. Integration by parts can be applied multiple times if needed, especially in cases where the resulting integral simplifies with repeated application.
4. The LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) helps in choosing \(u\).
5. It is useful for integrals involving products of polynomial and exponential or trigonometric functions.

Review Questions

Related terms

Product Rule:

A basic rule in differentiation that states: $\frac{d}{dx} [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$.

Substitution Method:

Another technique used for integration where a substitution is made to simplify the integral.

Definite Integral:

$$\int_{a}^{b} f(x) \, dx$$ represents the signed area under a curve between limits \(a\) and \(b\).