Problem 1 - Calculus Early Transcendentals

Question

On which derivative rule is integration by parts based?

Step-by-Step Solution

Answer

Answer: Integration by parts is based on the Product Rule for derivatives which states that the derivative of a product of two functions, u(x) and v(x), is given by: \( (u(x)v(x))' = u'(x)v(x) + u(x)v'(x) \). By rearranging this rule in terms of integration, we get the integration by parts formula: \( \int u'(x)v(x) dx = u(x)v(x) - \int u(x)v'(x) dx \).

Step 1: Identify the Derivative Rule

Integration by parts is based on the Product Rule for derivatives.

Step 2: Recall the Product Rule

The Product Rule states that the derivative of a product of two functions, say u(x) and v(x), can be given by: \( (u(x)v(x))' = u'(x)v(x) + u(x)v'(x) \).

Step 3: Rearrange the Product Rule to Formulate Integration by Parts

To rearrange the Product Rule in terms of integration, integrate both sides of the equation with respect to x: \( \int(u(x)v(x))' dx = \int(u'(x)v(x) + u(x)v'(x)) dx \). This results in: \( u(x)v(x) = \int u'(x)v(x) dx + \int u(x)v'(x) dx \).

Step 4: Arrive at the Integration by Parts Formula

By rearranging the terms, we get the Integration by Parts formula: \( \int u'(x)v(x) dx = u(x)v(x) - \int u(x)v'(x) dx \). Therefore, the integration by parts is based on the Product Rule for derivatives.