Substitution

Integration by substitution — often called u-substitution — is one of the most fundamental techniques for evaluating integrals that cannot be solved by simple antidifferentiation rules alone. The core idea is to replace a complicated expression inside an integral with a single new variable, transforming the integral into a simpler form that can be evaluated directly. It is the integration counterpart of the chain rule for differentiation, and mastering it unlocks a vast class of integrals that would otherwise seem intractable. The technique works by identifying a part of the integrand (the expression being integrated) that, along with its derivative, appears in the integral. By substituting u for that part and expressing du in terms of dx, the entire integral transforms into one written purely in terms of u. After evaluating the simpler integral, you substitute back to express the answer in terms of the original variable. This two-direction process — substituting in and then substituting back — is the hallmark of the method. Substitution is applicable to both indefinite and definite integrals. For definite integrals, there is an important choice: either convert the limits of integration into u-values and never substitute back, or keep the original limits and substitute back at the end. Both approaches give the same numerical answer when done correctly. Understanding which substitution to choose — and recognising when substitution is the right technique at all — is a skill developed through extensive practice.

Key takeaways
  • Identify the inner function
  • Compute du
  • Rewrite the integral in terms of u
  • Evaluate the u-integral
  • Substitute back (indefinite integrals)

Identify the inner function

Look at the integrand and identify a function u = g(x) whose derivative g'(x) also appears (possibly up to a constant multiple) as a factor in the integrand. A good candidate is often something inside a composite function — for example, inside a square root, an exponent, or a trigonometric function.

Compute du

Differentiate your chosen substitution: du/dx = g'(x), which we write as du = g'(x) dx. This step converts the differential dx into du. If g'(x) is off by a constant factor, you can compensate by multiplying or dividing both sides by that constant.

Rewrite the integral in terms of u

Substitute u for g(x) and replace every occurrence of g'(x) dx with du (adjusting for any constant factors). The resulting integral should be entirely in terms of u — no x should remain. If x still appears, you may need a different substitution or a different technique.

Evaluate the u-integral

Integrate with respect to u using standard antidifferentiation rules. For example, ∫u^n du = u^(n+1)/(n+1) + C, ∫e^u du = e^u + C, ∫cos(u) du = sin(u) + C, and so on.

Substitute back (indefinite integrals)

Replace u with g(x) to write the answer in terms of the original variable x, and include the constant of integration C. For definite integrals, either change the limits when you substitute (lower limit becomes g(a), upper limit becomes g(b)) and evaluate without substituting back, or keep original limits and substitute back before evaluating.

The Key Equation

∫f(g(x)) g′(x) dx=∫f(u) duwhere u=g(x)\int f(g(x))\, g'(x)\, dx = \int f(u)\, du \quad \text{where } u = g(x)
  • u — substitution variable (dimensionless): The new variable introduced to simplify the integral; defined as u = g(x), an inner function of x.
  • g'(x) — derivative of the inner function (dimensionless): The derivative of the substituted expression g(x) with respect to x. It must appear as a factor in the integrand for the substitution to work cleanly.
  • f(u) — outer function in terms of u (dimensionless): The simplified integrand after substitution; this should be a standard form that is easy to integrate directly.
  • dx,\, du — differentials (dimensionless): The infinitesimal elements with respect to which integration is performed. They are related by du = g'(x) dx.

Worked Example

Problem. Evaluate the indefinite integral ∫ 2x · cos(x²) dx.

sin⁡(x2)+C\sin(x^2) + C

Answer. sin(x²) + C

Figure
Diagram
u-Substitution: Transforming the IntegralOriginal Integral∫ 2x · cos(x²) dxLet u = x²Transformed Integral∫ cos(u) duStandard formdu = 2x dxAfter Integrationsin(u) + CintegrateFinal Answersin(x²) + Cu → x²
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
GW
Gottfried Wilhelm Leibniz
1646–1716 · German

Leibniz independently developed calculus in the 1670s and introduced the notation we use today: ∫ for the integral sign (an elongated 'S' for 'summa') and dx for the differential. His flexible notation — in particular treating du and dx as manipulable quantities — is precisely what makes u-substitution feel natural and systematic. The formula du = g'(x) dx is a direct expression of Leibniz's differential notation. Leibniz chose the integral sign ∫ in 1675. He also invented a mechanical calculating machine capable of multiplication and division, predating modern computers by centuries.

  • The Chain Rule — u-substitution is the integration mirror of the chain rule. Understanding how composite functions are differentiated makes it clear why substitution works and how to choose u.
  • Antiderivatives and Indefinite Integrals — Substitution is applied after setting up an indefinite integral; fluency with basic antiderivative rules (power rule, trig, exponential) is essential for evaluating the simplified u-integral.
  • Definite Integrals and the Fundamental Theorem — When substitution is applied to definite integrals, the limits of integration must be converted or properly handled — a key extension of the basic technique.
  • Integration by Parts — Integration by parts is the other major algebraic integration technique. Knowing when to use substitution versus integration by parts is a central skill in integral calculus.