Integration by Parts- Equation, Steps, and Examples
What Is Integration by Parts?
Integration by parts is a technique that breaks down complicated integrals into simpler pieces. It's the reverse of the product rule for differentiation. If you have an integral that looks impossible, this method often unlocks it.
The catch? You have to pick which part goes where—and picking wrong means starting over. That's the part textbooks skip.
The Formula
Here's the equation:
∫ u dv = uv - ∫ v du
That's it. Two functions, one subtracted integral. The entire method revolves around choosing u and dv correctly.
The derivative of u gives you du. The antiderivative of dv gives you v. Then you plug into the formula and hope the new integral is easier.
When to Use Integration by Parts
Not every integral needs this method. Use it when you have:
- A product of two functions
- Powers of x multiplied by exponential, logarithmic, or trigonometric functions
- Integrals that resist substitution
The LIATE Rule
This helps you pick u. Higher on the list gets priority:
- L - Logarithmic functions (ln x, log x)
- I - Inverse trigonometric functions (arcsin x, arctan x)
- A - Algebraic functions (x², 3x + 1)
- T - Trigonometric functions (sin x, cos x)
- E - Exponential functions (eˣ, 2ˣ)
Example: ∫ x·ln(x) dx → u = ln(x) because L comes before A.
Step-by-Step Process
Follow this order every time:
- Identify the two factors in your integral
- Choose u using LIATE, let dv be everything else
- Find du by differentiating u
- Find v by integrating dv
- Apply the formula: uv - ∫v du
- Solve the remaining integral
Sometimes you need to apply integration by parts twice. Sometimes you end up with the original integral—which means you solve algebraically.
Examples
Example 1: ∫ x·eˣ dx
Step 1: Choose u = x (algebraic) and dv = eˣ dx (exponential)
Step 2: du = dx, v = eˣ
Step 3: Apply formula:
∫ x·eˣ dx = x·eˣ - ∫ eˣ dx
Step 4: Solve remaining integral:
= x·eˣ - eˣ + C
Step 5: Factor if possible:
= eˣ(x - 1) + C
Example 2: ∫ x²·sin(x) dx
This one needs integration by parts twice.
First pass: u = x², dv = sin(x)dx
= -x²·cos(x) + ∫ 2x·cos(x) dx
Second pass: For ∫ 2x·cos(x) dx, use u = 2x, dv = cos(x)dx
= -x²·cos(x) + 2x·sin(x) + 2cos(x) + C
Messy, but solvable.
Example 3: ∫ ln(x) dx
This looks like one function, but ln(x) is actually 1·ln(x). Set dv = dx.
u = ln(x), dv = dx
du = 1/x dx, v = x
= x·ln(x) - ∫ x·(1/x) dx
= x·ln(x) - ∫ 1 dx
= x·ln(x) - x + C
Common Mistakes to Avoid
- Forgetting the constant of integration. Always add +C at the end.
- Wrong u choice. If your new integral looks worse, try swapping u and dv.
- Sign errors. The formula is uv - ∫v du. The minus sign is not optional.
- Dropping terms. When applying the formula, don't forget to include the uv part.
Integration by Parts vs. Other Methods
| Method | Best For | Difficulty |
|---|---|---|
| Substitution | Chain rule reversed | Easier |
| Integration by Parts | Products with logs, trig, exponentials | Medium |
| Trig Substitution | Squares under radicals | Harder |
| Partial Fractions | Rational functions with distinct factors | Medium |
If substitution fails, try parts. If parts gets messy, maybe trig substitution or partial fractions fits better.
Practice Tips
- Memorize the formula. Write it 10 times until it's automatic.
- Always check your answer by differentiating it. You should get the original integrand.
- Start with simple examples before tackling integrals with three or more factors.
- Keep track of your signs. Negative signs compound quickly.
Integration by parts is a tool. Like any tool, you get better by using it. The formula is simple—the hard part is recognizing when it applies and choosing your u and dv correctly.