1. Why Integration by Parts Is Needed
Some integrals involve a product of functions where substitution does not simplify the expression. Integration by parts helps break such products into simpler integrals. In personal-notes style: when a function is too tangled to integrate directly, parts helps separate it cleanly.
2. Idea Behind Integration by Parts
The rule comes from reversing the product rule of differentiation. By rearranging the product rule, we turn the integral of a product into a combination of simpler terms.
2.1. Derivation from Product Rule
Start from the product rule:
\frac{d}{dx}(u \cdot v) = u'v + uv'
Integrating both sides gives the basic formula.
3. Formula for Integration by Parts
\int u \, dv = uv - \int v \, du
Here, u and dv are chosen from the original integrand.
4. How to Choose u and dv
Choosing u and dv correctly makes the process smoother. A common guideline is LIATE.
4.1. LIATE Rule
- Logarithmic
- Inverse trigonometric
- Algebraic
- Trigonometric
- Exponential
Choose u from the top of the list when possible.
5. Step-by-Step Method
- Pick u and dv from the integrand.
- Differentiate u to find du.
- Integrate dv to find v.
- Apply the formula: \( uv - \int v \, du \).
- Simplify the resulting integral.
6. Examples
These examples show how integration by parts simplifies different combinations of functions.
6.1. Example 1 — Polynomial × Exponential
Evaluate:
\int x e^x dx
Choose:
- u = x → du = dx
- dv = e^x dx → v = e^x
Apply the formula:
\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C
6.2. Example 2 — Polynomial × Trig
Evaluate:
\int x \sin x dx
Choose:
- u = x → du = dx
- dv = \sin x dx → v = -\cos x
Apply the formula:
\int x \sin x dx = -x \cos x + \int \cos x dx = -x \cos x + \sin x + C
6.3. Example 3 — Integral of ln(x)
Evaluate:
\int \ln x dx
We take:
- u = ln x (logarithmic → top priority)
- dv = dx
Then:
du = \frac{1}{x} dx, \quad v = x
Apply the formula:
\int \ln x dx = x \ln x - x + C
7. When Integration by Parts Works Best
Integration by parts is most effective when:
- One function becomes simpler when differentiated.
- The other function is easy to integrate.
- The remaining integral is easier than the original.