Integration by Substitution: u-Substitution Method Guide

The chain rule lets us differentiate a composite function like (x² + 1)⁵, but reversing that process looks impossible at first: the derivative has multiplied an extra 2x into the expression. Integration by substitution (also called u-substitution) is the trick that reverses the chain rule. You introduce a new variable u for the inner function, so the messy composite collapses into a simple standard integral.

This method matters because a huge share of exam integrals are chain-rule reversals in disguise. Whenever you see a composite function multiplied by something that looks like the derivative of its inner part, substitution is almost certainly the intended method, and spotting that pattern saves minutes of trial and error.

Key Concepts

• Substitution: set u = g(x), the inner function of the composite.
• Differential swap: du = g′(x) dx, so g′(x) dx is replaced by du everywhere.
• The key signal: the integrand contains both a composite f(g(x)) and the factor g′(x) (possibly with a constant multiple).
• Back-substitution: after integrating in u, replace u with g(x) in the final answer.
• Constant adjustment: if the integrand has k·g′(x) instead of g′(x), pull the constant factor k out first.

Formulas

• ∫ f(g(x)) · g′(x) dx = ∫ f(u) du, where u = g(x)
• ∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / [a(n+1)] + C
• ∫ f′(x)/f(x) dx = ln|f(x)| + C (a special, very common case)
• ∫ e^(g(x)) · g′(x) dx = e^(g(x)) + C

Worked Example

Evaluate ∫ 2x (x² + 1)⁵ dx. Step 1: the inner function is x² + 1, whose derivative 2x is sitting right beside it — perfect signal. Set u = x² + 1, so du = 2x dx. Step 2: replace everything: the integral becomes ∫ u⁵ du. Step 3: integrate using the power rule to get u⁶/6 + C. Step 4: back-substitute: (x² + 1)⁶/6 + C. Differentiating the answer with the chain rule returns 2x(x² + 1)⁵, which confirms the result.

Common Misconceptions

• Substituting but leaving dx behind: every x must disappear after substitution. If an x survives, your chosen u is probably wrong.
• Skipping back-substitution: an answer in terms of u is incomplete for an indefinite integral; always return to x.
• Forcing substitution on everything: ∫ x² sin x dx has no inner-function pair — it needs integration by parts, not substitution.

Real-World Applications

Substitution appears whenever a physical quantity depends on a composite expression, such as the velocity of an object under a force that depends on position through energy terms. In probability, it converts messy density functions into standard normal forms. In engineering, it simplifies integrals of damped oscillations and decaying signals into textbook cases.

Integration by substitution is simply the chain rule read backwards. Train your eye to spot the inner function and its derivative side by side, and a large class of integrals reduces to a standard formula in three short steps.