u-Substitution Formula
The Formula
For definite integrals: change the bounds too! If u = g(x), then when x = a, u = g(a).
When to use: When you see a composite function inside an integral along with its inner derivative lurking nearby, substitution collapses the composition into a single variable. It's like un-nesting a function: replace the inner part with u, and the integral becomes simpler.
Quick Example
= \int \cos u\,du = \sin u + C = \sin(x^2) + C
Notation
What This Formula Means
An integration technique where you substitute u = g(x) and du = g'(x)\,dx to transform a complicated integral into a simpler one. It is the reverse of the chain rule for differentiation.
When you see a composite function inside an integral along with its inner derivative lurking nearby, substitution collapses the composition into a single variable. It's like un-nesting a function: replace the inner part with u, and the integral becomes simpler.
Formal View
Worked Examples
Example 1
easySolution
- 1 Let u = x^3+1, so du = 3x^2\,dx.
- 2 Integral becomes \int u^4\,du = \frac{u^5}{5} + C.
- 3 Substitute back: \frac{(x^3+1)^5}{5} + C.
Answer
Example 2
mediumCommon Mistakes
- Forgetting to change dx to du: after substituting u = g(x), you MUST replace dx with \frac{du}{g'(x)}. Leaving a mix of u and dx is meaningless.
- Not changing the limits of integration for definite integrals: if u = x^2 and x goes from 0 to 3, then u goes from 0 to 9. Either change the bounds OR substitute back to x before evaluating.
- Choosing u poorly: if your substitution makes the integral MORE complicated, try a different choice. A good u makes du appear (up to a constant) in the remaining integrand.
Why This Formula Matters
This is the single most important integration technique—it's the first method to try on any non-trivial integral. Most integrals in practice require at least one substitution.
Frequently Asked Questions
What is the u-Substitution formula?
An integration technique where you substitute u = g(x) and du = g'(x)\,dx to transform a complicated integral into a simpler one. It is the reverse of the chain rule for differentiation.
How do you use the u-Substitution formula?
When you see a composite function inside an integral along with its inner derivative lurking nearby, substitution collapses the composition into a single variable. It's like un-nesting a function: replace the inner part with u, and the integral becomes simpler.
What do the symbols mean in the u-Substitution formula?
Let u = g(x), then du = g'(x)\,dx. After integrating in terms of u, substitute back to express the result in terms of x.
Why is the u-Substitution formula important in Math?
This is the single most important integration technique—it's the first method to try on any non-trivial integral. Most integrals in practice require at least one substitution.
What do students get wrong about u-Substitution?
The trickiest part is choosing the right u. Look for a function whose derivative also appears in the integrand (perhaps off by a constant factor). The 'inner function' of a composition is usually a good first guess.
What should I learn before the u-Substitution formula?
Before studying the u-Substitution formula, you should understand: integral, chain rule.
Want the Full Guide?
This formula is covered in depth in our complete guide:
How to Integrate Rational Functions: Long Division and Partial Fractions →