IntegrationByParts - crowlogic/arb4j GitHub Wiki

Integration by parts is a technique used in calculus to compute the integral of a product of two functions. It is based on the product rule for differentiation and can be seen as the "reverse" of the product rule. The formula for integration by parts is derived from the product rule, and it states:

$$\int u(x) dv(x) = u(x)v(x) - \int v(x) du(x)$$

where $u(x)$ and $v(x)$ are differentiable functions, and $du(x)$ and $dv(x)$ are their respective derivatives with respect to the variable of integration (usually $x$). In practice, you need to choose $u(x)$ and $dv(x)$ from the product of the two functions in the given integral, then differentiate $u(x)$ to find $du(x)$ and integrate $dv(x)$ to find $v(x)$.

To apply integration by parts, follow these steps:

1. Identify the two functions in the product that you want to integrate, and assign one to $u(x)$ and the other to $dv(x)$.
2. Differentiate $u(x)$ to find $du(x)$ and integrate $dv(x)$ to find $v(x)$.
3. Substitute the expressions for $u(x)$, $v(x)$, and $du(x)$ into the integration by parts formula.
4. Evaluate the remaining integral (if possible) and combine the results to find the final solution.

Integration by parts can be applied multiple times, especially when dealing with products of functions that require several iterations to simplify the integral. Sometimes, it can also be combined with other integration techniques, like substitution or trigonometric identities, to solve more complex integrals.