Substitution Rule
Reverse of the chain rule.
Key idea
Reverse of the chain rule.
In depth
Substitution reverses the chain rule. Let u = g(x), then du = g′(x) dx, and rewrite the entire integral in u, including limits for definite integrals.
Worked example
∫ 2x cos(x²) dx: u = x², du = 2x dx → ∫ cos u du = sin(x²) + C.
Formula
Real-world uses
- Integrating formulas for population and signal models after a change of variable
Common mistake
Not converting every x or dx into u terms, or forgetting to change the limits.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Substitution Rule?
Reverse of the chain rule. It is a topic at the expert level in Calculus.
What is the formula for Substitution Rule?
The key formula is ∫ f(g(x))g′(x)dx = ∫ f(u)du.
Where is Substitution Rule used in real life?
Real-world examples: Integrating formulas for population and signal models after a change of variable.
What should I learn before Substitution Rule?
Make sure you are comfortable with Properties of Definite Integrals, the previous topic in Calculus.
What is a common mistake with Substitution Rule?
Not converting every x or dx into u terms, or forgetting to change the limits.
How can I practise Substitution Rule?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Substitution Rule?
Move on to Integration by Parts next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.