What is the integration by parts rule for definite integrals?

The integration by parts rule for definite integrals is a technique used to solve integrals that involve the product of two functions. It's a powerful tool for simplifying complex integrals and is a direct extension of the integration by parts rule for indefinite integrals.

Understanding the Rule

The formula for integration by parts for definite integrals is:

∫[a, b] u(x)v'(x) dx = [u(x)v(x)]_[a, b] - ∫[a, b] u'(x)v(x) dx

Where:

• u(x) and v(x) are differentiable functions of x.
• u'(x) and v'(x) are their respective derivatives.
• [a, b] represents the interval of integration.
• [u(x)v(x)]_[a, b] denotes the evaluation of the product u(x)v(x) at the upper limit (b) minus the evaluation at the lower limit (a).

Practical Application

The key to effectively using integration by parts is choosing the right functions for u(x) and v(x). The goal is to simplify the second integral on the right-hand side of the equation, making it easier to solve.

Example:

Let's calculate the definite integral of *x sin(x) from 0 to π**:

1. Choose u(x) and v'(x): Let u(x) = x and v'(x) = sin(x).

2. Find u'(x) and v(x): u'(x) = 1 and v(x) = -cos(x).

3. Apply the formula:

   ∫[0, π] x * sin(x) dx = [-x * cos(x)]_[0, π] - ∫[0, π] -cos(x) dx

4. Evaluate the first term and solve the second integral:

   = [(-π * cos(π)) - (0 * cos(0))] + ∫[0, π] cos(x) dx
   = π + [sin(x)]_[0, π]
   = π + (sin(π) - sin(0))
   = π

Therefore, the definite integral of *x sin(x) from 0 to π is π**.

Key Points

• The integration by parts rule is a powerful tool for solving definite integrals involving products of functions.
• It is essential to choose suitable functions for u(x) and v(x) to simplify the integral.
• The rule is a direct extension of the integration by parts rule for indefinite integrals.