January 20, 2025 • 10 min read

U-Substitution Step by Step: From Confusion to Clarity

U-substitution is the most important integration technique you'll learn. Once it clicks, you'll see it everywhere. Let's break it down completely.

What is U-Substitution?

U-substitution is the reverse of the chain rule. When we differentiate f(g(x)), we get f'(g(x))·g'(x). U-substitution reverses this process.

The Core Idea

$$\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du = F(u) + C$$

Where u = g(x) and du = g'(x)dx

The 5-Step Method

  1. Identify the inner function — What's "inside" another function?
  2. Let u = inner function
  3. Find du — Differentiate u with respect to x
  4. Rewrite the integral in terms of u — Everything must be in u
  5. Integrate and substitute back

Example 1: Basic Pattern

$$\int 2x \cdot \cos(x^2) \, dx$$

Step 1: The inner function is x² (inside the cosine).

Step 2: Let u = x²

Step 3: du/dx = 2x, so du = 2x dx

Step 4: Substitute: ∫ cos(u) du

Step 5: = sin(u) + C = sin(x²) + C

Example 2: Adjusting for Constants

$$\int x \cdot e^{x^2} \, dx$$

Notice: If we let u = x², then du = 2x dx. But we only have x dx, not 2x dx.

Solution: Multiply and divide by 2:

  • u = x², du = 2x dx, so x dx = (1/2) du
  • ∫ eᵘ · (1/2) du = (1/2) ∫ eᵘ du = (1/2)eᵘ + C
  • Answer: (1/2)e^(x²) + C

Example 3: Trig Functions

$$\int \sin^3(x) \cos(x) \, dx$$

Key insight: cos(x) is the derivative of sin(x)!

  • Let u = sin(x), then du = cos(x) dx
  • ∫ u³ du = u⁴/4 + C
  • Answer: sin⁴(x)/4 + C

Example 4: Definite Integrals

$$\int_0^1 x(x^2 + 1)^4 \, dx$$

Two approaches:

Method A: Change the Limits

  • u = x² + 1, du = 2x dx → x dx = (1/2) du
  • When x = 0: u = 1
  • When x = 1: u = 2
  • ∫₁² (1/2)u⁴ du = (1/2)·[u⁵/5]₁² = (1/10)(32-1) = 31/10

Method B: Substitute Back First

  • ∫ (1/2)u⁴ du = u⁵/10 + C = (x²+1)⁵/10 + C
  • [(x²+1)⁵/10]₀¹ = (2)⁵/10 - (1)⁵/10 = 32/10 - 1/10 = 31/10

Example 5: Logarithmic

$$\int \frac{\ln(x)}{x} \, dx$$

The derivative of ln(x) is 1/x, which is right there!

  • Let u = ln(x), du = (1/x) dx
  • ∫ u du = u²/2 + C
  • Answer: [ln(x)]²/2 + C

Example 6: Square Roots

$$\int \frac{x}{\sqrt{1 + x^2}} \, dx$$
  • Let u = 1 + x², du = 2x dx, so x dx = (1/2) du
  • ∫ (1/2) u^(-1/2) du = (1/2) · 2u^(1/2) + C = √u + C
  • Answer: √(1 + x²) + C

Example 7: Tangent and Secant

$$\int \tan(x) \, dx$$

Write tan(x) = sin(x)/cos(x):

  • Let u = cos(x), du = -sin(x) dx
  • ∫ -1/u du = -ln|u| + C
  • Answer: -ln|cos(x)| + C = ln|sec(x)| + C

Pro Tips for U-Substitution

How to Spot Substitution Opportunities:

  • Look for nested functions — something inside something else
  • Check if the derivative is present — or a constant multiple of it
  • Watch for typical patterns: f(x)·f'(x), e^(g(x))·g'(x), [g(x)]ⁿ·g'(x)

Common Mistakes to Avoid:

  • ❌ Forgetting to substitute back to x at the end
  • ❌ Not adjusting for constant multipliers (2, 3, 1/2, etc.)
  • ❌ Leaving x terms in the integral after substitution
  • ❌ Forgetting to change limits for definite integrals

Practice These

Try these on your own, then check the solutions page:

  1. ∫ 3x²·(x³+1)⁵ dx
  2. ∫ cos(x)·e^(sin x) dx
  3. ∫ x/(x²+4) dx
  4. ∫ sec²(x)·tan(x) dx
More Practice Learn Integration by Parts