January 18, 2025 • 8 min read

Integration by Parts: 15 Worked Examples

Integration by parts handles products that substitution can't. Here are 15 examples organized by difficulty, from basic to challenging.

The Formula

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

The LIATE Rule

Choose u from this list (first = highest priority):

  • Logarithmic: ln(x), log(x)
  • Inverse trig: arcsin(x), arctan(x)
  • Algebraic: x, x², polynomials
  • Trigonometric: sin(x), cos(x)
  • Exponential: eˣ, 2ˣ

Basic Examples

Example 1
$$\int x \cdot e^x \, dx$$

Solution: u = x, dv = eˣ dx → du = dx, v = eˣ

$$= x \cdot e^x - \int e^x \, dx = xe^x - e^x + C = e^x(x-1) + C$$
Example 2
$$\int x \cdot \cos(x) \, dx$$

Solution: u = x, dv = cos(x) dx → du = dx, v = sin(x)

$$= x\sin(x) - \int \sin(x) \, dx = x\sin(x) + \cos(x) + C$$
Example 3
$$\int \ln(x) \, dx$$

Solution: u = ln(x), dv = dx → du = (1/x) dx, v = x

$$= x\ln(x) - \int x \cdot \frac{1}{x} \, dx = x\ln(x) - x + C$$

Medium Examples

Example 4
$$\int x^2 \cdot e^x \, dx$$

Solution: Apply by parts twice!

First: u = x², dv = eˣ dx → x²eˣ - 2∫xeˣ dx

Second: Apply Example 1 result

$$= x^2 e^x - 2(xe^x - e^x) + C = e^x(x^2 - 2x + 2) + C$$
Example 5
$$\int x^2 \cdot \sin(x) \, dx$$

Solution: Apply twice with u = x² first, then u = 2x

$$= -x^2\cos(x) + 2x\sin(x) + 2\cos(x) + C$$
Example 6
$$\int \arctan(x) \, dx$$

Solution: u = arctan(x), dv = dx

$$= x\arctan(x) - \int \frac{x}{1+x^2} \, dx = x\arctan(x) - \frac{1}{2}\ln(1+x^2) + C$$

Cyclic Examples (Bounce-Back)

Example 7
$$\int e^x \cdot \sin(x) \, dx$$

Solution: The integral reappears! Let I = ∫eˣsin(x)dx

  • First: u = sin(x), dv = eˣ dx → I = eˣsin(x) - ∫eˣcos(x)dx
  • Second: u = cos(x), dv = eˣ dx → I = eˣsin(x) - [eˣcos(x) + ∫eˣsin(x)dx]
  • I = eˣsin(x) - eˣcos(x) - I
  • 2I = eˣ(sin(x) - cos(x))
$$= \frac{e^x(\sin(x) - \cos(x))}{2} + C$$
Example 8
$$\int e^x \cdot \cos(x) \, dx$$

Solution: Similar cyclic approach

$$= \frac{e^x(\sin(x) + \cos(x))}{2} + C$$

Definite Integral Examples

Example 9
$$\int_0^1 x \cdot e^{2x} \, dx$$

Solution: u = x, dv = e^(2x) dx → v = e^(2x)/2

$$= \left[\frac{xe^{2x}}{2}\right]_0^1 - \frac{1}{2}\int_0^1 e^{2x} \, dx = \frac{e^2}{2} - \frac{1}{4}(e^2 - 1) = \frac{e^2 + 1}{4}$$

Tabular Method (DI Method)

For repeated integration by parts with polynomial × exponential/trig:

Steps:

  1. Make two columns: D (derivatives) and I (integrals)
  2. Put polynomial in D column, other function in I column
  3. Differentiate D column until you get 0
  4. Integrate I column the same number of times
  5. Multiply diagonally with alternating signs (+, -, +, -...)
Example 10
$$\int x^3 e^x \, dx$$

Tabular Method:

D: x³ → 3x² → 6x → 6 → 0

I: eˣ → eˣ → eˣ → eˣ → eˣ

$$= e^x(x^3 - 3x^2 + 6x - 6) + C$$

More Examples

Example 11
$$\int x \cdot \ln(x) \, dx$$
$$= \frac{x^2 \ln(x)}{2} - \frac{x^2}{4} + C$$
Example 12
$$\int (\ln x)^2 \, dx$$
$$= x(\ln x)^2 - 2x\ln(x) + 2x + C$$
Example 13
$$\int x \cdot \arcsin(x) \, dx$$
$$= \frac{x^2 \arcsin(x)}{2} + \frac{x\sqrt{1-x^2}}{4} - \frac{\arcsin(x)}{4} + C$$

Key Tips

  • ✅ Use LIATE to choose u (logarithms beat exponentials)
  • ✅ If your integral comes back, solve algebraically!
  • ✅ Consider tabular method for polynomials
  • ✅ Sometimes you need to apply by parts multiple times
  • ❌ Don't forget to integrate dv correctly
Practice More Common Mistakes