Calculus II: Integration by Parts

Integration by parts is more than a formula—it's an essential problem-solving strategy that transforms complex integrals into simpler ones, a skill indispensable for engineering fields like dynamics, signal processing, and thermodynamics. Mastering it requires understanding its conceptual origin, developing a reliable method for selecting parts, and knowing when to apply it versus other techniques. This guide provides a systematic, engineer-focused approach to wielding this powerful tool effectively.

The Foundation: Derivation from the Product Rule

The formula for Integration by Parts is derived directly from the Product Rule for differentiation. If you recall, the derivative of a product of two functions $u(x)$ and $v(x)$ is given by $\frac{d}{dx}[u(x)v(x)]=u^{\prime }v+u{v}^{\prime }$. If we rearrange this and integrate both sides with respect to $x$, we arrive at the core formula:

$$\int udv=uv-\int vdu$$

This formula swaps one integral, $\int udv$, for another, $\int vdu$. The entire strategy hinges on making a wise initial choice for $u$ and $dv$ so that the new integral on the right-hand side is simpler than the one you started with. This is not just algebraic manipulation; it's a tactical exchange, trading the current difficult integral for a (hopefully) easier one to solve.

The LIATE Guideline for Choosing u and dv

Choosing $u$ and $dv$ is the critical decision point. A poor choice can lead to an integral more complicated than the original. The LIATE guideline (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) provides a reliable hierarchy for selecting $u$. You should let $u$ be the function in your integrand that comes first in the LIATE list.

For example, consider $\int x{e}^{x}dx$. The integrand contains an Algebraic function ($x$) and an Exponential function ($e^x$). According to LIATE, Algebraic (A) comes before Exponential (E). Therefore, we choose:

• $u=x$ (Algebraic)
• $dv=e^{x}dx$ (the rest, which is Exponential)

We then differentiate $u$ to find $du=dx$ and integrate $dv$ to find $v=e^x$. Applying the formula: $$\int x{e}^{x}dx=x{e}^x-\int e^{x}dx=x{e}^x-e^x+C$$ The new integral $\int e^{x}dx$ is trivial, confirming our choice was correct. LIATE is a heuristic, not an absolute law, but it succeeds in the vast majority of cases you will encounter in engineering calculus.

The Tabular Integration Method

For integrals requiring repeated integration by parts, such as $\int x^3\sin (x)dx$, the tabular method (also called the "DI method") organizes the work efficiently. It is particularly useful when differentiating one part of the product ($u$) eventually reduces to zero.

The process involves creating two columns. Repeatedly differentiate the $u$ column and repeatedly integrate the $dv$ column. Then, combine the terms diagonally, alternating signs starting with "+".

For $\int x^3\sin (x)dx$:

Multiply diagonally and sum with the alternating signs: $$\int x^3\sin (x)dx=(+)(x^3)(-\cos (x))+(-)(3x^2)(-\sin (x))+(+)(6x)(\cos (x))+(-)(6)(\sin (x))+C$$ Which simplifies to: $$=-x^3\cos (x)+3x^2\sin (x)+6x\cos (x)-6\sin (x)+C$$ This method streamlines repeated applications, saving time and reducing algebraic errors.

Applications and Strategic Recognition

Integration by parts has direct applications, especially with definite integrals. When applying the formula ${\int }_a^{b}udv=[uv{]}_a^b-{\int }_a^{b}vdu$, you must evaluate the $uv$ term at the bounds $a$ and $b$ as the first step. This is common in engineering probability and Fourier analysis.

Recognizing integrands that require this technique is a key skill. You should suspect integration by parts when the integrand is a product of two distinct function types (e.g., polynomial × exponential, polynomial × trigonometric, logarithmic × algebraic). Importantly, you must also recognize when not to use it. If an integrand resembles a standard form from a basic substitution (like $\int e^{x^2}xdx$, which is a simple $u$-sub with $u=x^2$), that simpler method should be used first. Integration by parts is often the tool for integrals that do not yield to more straightforward techniques.

Common Pitfalls

1. Misapplying LIATE for Inverse Trig Functions: A common error is mishandling integrals like $\int \ln (x)dx$. This integrand appears to be a single function, but you can view it as $1\cdot \ln (x)$. Here, $u=\ln (x)$ (Logarithmic, top of LIATE) and $dv=1dx$. Forgetting that "1" is a valid choice for $dv$ can leave you stuck.

1. Ignoring the Constant of Integration When Finding v: When you integrate $dv$ to find $v$, you technically get a constant of integration: $v+C$. However, in the standard integration by parts formula, you should always use $v$ without the constant. Choosing any non-zero constant will cancel itself out in the final result, but it makes the intermediate work unnecessarily messy. Simply use the simplest antiderivative (where $C=0$).

1. Giving Up Too Soon on Repeated Applications: When faced with an integral like $\int e^x\cos (x)dx$, applying integration by parts once often leads to a similar, unsolved integral. The strategic move is to apply the technique a second time, which frequently produces an equation you can solve for the original integral. Stopping after one cycle misses this elegant resolution.

1. Forgetting to Evaluate the uv Term in Definite Integrals: In the rush to compute the new integral $\int vdu$, it's easy to forget the evaluated boundary term $[uv{]}_a^b$. Always write this term explicitly and compute its value before proceeding.

Summary

• Integration by Parts, derived from the Product Rule, is a strategic exchange: $\int udv=uv-\int vdu$.
• The LIATE guideline (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) is a reliable heuristic for choosing $u$ to simplify the resulting integral $\int vdu$.
• The tabular method efficiently handles integrals requiring repeated application, such as polynomials times sine/cosine/exponential functions.
• For definite integrals, remember to compute the boundary term $[uv{]}_a^b$ as part of the formula.
• Recognize the technique's niche: it is primary for products of different function types where simpler substitution fails, and it is often the first step for integrating logarithmic and inverse trigonometric functions.