A definite integral measures a signed accumulation over an interval, such as area, displacement, or total change. Many definite integrals look complicated because one function is nested inside another. Substitution is a method for rewriting the integral in terms of a new variable so the structure becomes simpler.
It matters because it turns difficult integrals into familiar forms while preserving the meaning of the original accumulation.
Key Facts
- Definite integral notation: ∫[a,b] f(x) dx means accumulate f(x) from x = a to x = b.
- Substitution rule: if u = g(x), then du = g'(x) dx.
- For definite integrals, ∫[a,b] f(g(x))g'(x) dx = ∫[g(a),g(b)] f(u) du.
- Change the bounds when you change variables: lower bound becomes u = g(a), upper bound becomes u = g(b).
- Example: ∫[0,2] 2x(x^2 + 1)^3 dx, let u = x^2 + 1, du = 2x dx, bounds 1 to 5, so ∫[1,5] u^3 du.
- After changing bounds to u-values, do not substitute back to x before evaluating.
Vocabulary
- Definite integral
- A definite integral gives the signed accumulation of a function over a specific interval.
- Substitution
- Substitution is a technique that replaces a complicated expression with a simpler variable, usually u.
- Differential
- A differential such as du or dx represents how a variable changes inside an integral.
- Bounds of integration
- Bounds of integration are the lower and upper values that define the interval of a definite integral.
- Antiderivative
- An antiderivative of a function is another function whose derivative is the original function.
Common Mistakes to Avoid
- Keeping the old x-bounds after substituting u is wrong because the integral is now measured in the u-variable, so the endpoints must be converted.
- Forgetting to replace dx correctly is wrong because du must match the derivative of the substituted expression, including constants and signs.
- Substituting back to x after already changing the bounds is wrong because the integral can be evaluated completely in u once the limits are updated.
- Choosing u as the whole integrand is usually wrong because a useful substitution should reveal a derivative factor that pairs with dx.
Practice Questions
- 1 Evaluate ∫[0,3] 2x(x^2 + 4)^2 dx using u-substitution.
- 2 Evaluate ∫[1,4] (3/(2√(3x + 1))) dx using u = 3x + 1.
- 3 Explain why changing the limits from x-values to u-values lets you avoid substituting back to x at the end of a definite integral.