Learning Outcomes

Upon completing this lesson, you will be able to:

• Recognize when to use the technique of Integration by Parts.
• Apply the Integration by Parts formula to find the integrals of products of functions.
• Use the LIATE mnemonic to make a strategic choice for $u$ in the Integration by Parts formula.
• Use Tabular Integration as an efficient shortcut for repeated Integration by Parts.
• Solve a variety of integrals by choosing the appropriate technique ($u$-substitution or integration by parts).

Topics Covered

• Integration by Parts
• Tabular Integration
• Examples and Problem-Solving Strategies

I. Integration by Parts

While u-substitution is used to reverse the Chain Rule, Integration by Parts is the technique used to reverse the Product Rule. It is the primary method for finding the integral of a product of two functions.

A. Deriving the Formula

The Product Rule for differentiation states:

$$ \frac{d}{dx}[u(x)v(x)] = u(x)v'(x) + v(x)u'(x) $$

If we integrate both sides with respect to $x$, we get:

$$ u(x)v(x) = \int u(x)v'(x) ,dx + \int v(x)u'(x) ,dx $$

Rearranging this equation to solve for one of the integrals gives us the formula for Integration by Parts.

B. The Integration by Parts Formula

Let $u$ and $v$ be functions of $x$. The formula is:

$$ \int u ,dv = uv - \int v ,du $$

To use this formula, we split the original integrand into two parts: $u$ and $dv$.

1. We choose a part to be $u$. We will then need to differentiate it to find $du$.
2. The remaining part of the integrand (including $dx$) becomes $dv$. We will then need to integrate it to find $v$.

C. How to Choose $u$ - The LIATE Rule

The key to using Integration by Parts successfully is making a good choice for $u$. The goal is to choose a $u$ that becomes simpler when differentiated. A helpful mnemonic for choosing $u$ is LIATE:

• L - Logarithmic functions (e.g., $\ln(x)$, $\log_b(x)$)
• I - Inverse trigonometric functions (e.g., $\arcsin(x)$, $\arctan(x)$)