3.1 Integration by Parts

Integration by Parts

Integration by parts lets you handle integrals involving products of functions that don't yield to simpler methods like u-substitution. The core idea: it transforms one integral into another, hopefully easier one.

The formula comes directly from the product rule for derivatives, just rearranged. If you can differentiate one piece and integrate the other, this technique breaks the problem into manageable parts.

When to Use Integration by Parts

You'll reach for this technique when the integrand is a product of two functions where one simplifies when differentiated and the other is straightforward to integrate:

• Polynomial × trig or exponential: $\int x \sin(x)\, dx$, $\int x e^x\, dx$
• Polynomial × logarithm: $\int x \ln(x)\, dx$
• Polynomial × inverse trig: $\int x \arctan(x)\, dx$
• Logarithmic or inverse trig functions alone: $\int \ln(x)\, dx$ counts because you can treat it as $1 \cdot \ln(x)$

A good rule of thumb: if u-substitution and partial fractions don't work, and you see a product of "different types" of functions, try integration by parts.

The Formula and How to Apply It

The integration-by-parts formula is:

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

This comes from integrating both sides of the product rule $d(uv) = u\, dv + v\, du$ and rearranging.

Steps to apply it:

1. Choose $u$ and $dv$ from the integrand. Pick $u$ as the function that gets simpler when you differentiate it, and $dv$ as the rest (which you need to be able to integrate).

2. Differentiate $u$ to get $du$.

3. Integrate $dv$ to get $v$. You don't need a constant of integration here.

4. Plug everything into $uv - \int v\, du$.

5. Evaluate the new integral $\int v\, du$, which should be simpler than what you started with.

Choosing $u$ wisely is the most important step. The mnemonic LIATE ranks your choices from best to worst:

• Logarithmic ($\ln(x)$, $\log(x)$)
• Inverse trig ($\arctan(x)$, $\arcsin(x)$)
• Algebraic (polynomials like $x$, $x^2$)
• Trigonometric ($\sin(x)$, $\cos(x)$)
• Exponential ($e^x$, $2^x$)

Functions higher on this list make better choices for $u$ because they simplify when differentiated.

Repeated Integration by Parts

Some integrals require you to apply the formula more than once. This happens when the new integral $\int v\, du$ is still a product that needs integration by parts.

For $\int x^2 e^x\, dx$, the first application reduces $x^2$ to $x$, giving you $\int x e^x\, dx$, which itself requires a second round. Each application drops the polynomial degree by one.

There's also a special case with integrals like $\int e^x \sin(x)\, dx$. After applying integration by parts twice, you'll get the original integral appearing on the right side. Call the original integral $I$, set up the equation, and solve for $I$ algebraically.

Definite Integrals Using Integration by Parts

For definite integrals, the process is the same with one addition: you evaluate at the bounds.

1. Apply the integration-by-parts formula to get $uv - \int v\, du$.

2. Evaluate the $uv$ term at the upper and lower limits: $\Big[uv\Big]_a^b$.

3. Evaluate the remaining integral $\int_a^b v\, du$ using the same original limits.

4. Combine: $\Big[uv\Big]_a^b - \int_a^b v\, du$.

A common mistake is forgetting to apply the limits to the $uv$ term. The bounds apply to the entire expression, not just the leftover integral.

Connections to Other Techniques

Integration by parts often works alongside other methods. After one application, the remaining integral might call for u-substitution or partial fractions. Recognizing when to switch techniques is a skill that comes with practice.

This technique also appears frequently when solving differential equations and in deriving reduction formulas, where a general integral $\int \sin^n(x)\, dx$ gets expressed in terms of $\int \sin^{n-2}(x)\, dx$.