Integration by Substitution
Integration by substitution is a technique that simplifies complex integrals by replacing a part of the integrand with a new variable, effectively reversing the chain rule of differentiation. The method works by identifying an inner function u = g(x), computing du = g'(x) dx, and rewriting the integral entirely in terms of u. It is one of the most widely used integration techniques and is applicable whenever an integrand contains a composite function paired with its derivative.
Key Formula
∫ f(g(x)) · g'(x) dx = ∫ f(u) du, where u = g(x)
LaTeX: \int f(g(x))\, g'(x)\, dx = \int f(u)\, du, \quad u = g(x)
| Symbol | Meaning | Unit |
|---|---|---|
| u | Substitution variable, u = g(x) | dimensionless |
| g(x) | Inner function | dimensionless |
| g'(x) | Derivative of inner function | dimensionless |
| f(u) | Outer function expressed in u | dimensionless |
Worked Example
Problem
Evaluate ∫ 2x · cos(x²) dx.
Solution
Step 1: Identify inner function: let u = x², so du = 2x dx. Step 2: Rewrite the integral in terms of u: ∫ cos(x²) · 2x dx = ∫ cos(u) du. Step 3: Integrate: ∫ cos(u) du = sin(u) + C. Step 4: Back-substitute u = x²: sin(x²) + C.
Answer
∫ 2x · cos(x²) dx = sin(x²) + C
Integration by Substitution — Common Patterns
| Integral Pattern | Substitution | Result | Example | Key Clue |
|---|---|---|---|---|
| ∫f(ax+b) dx | u = ax+b | (1/a)F(u)+C | ∫sin(3x) dx = −cos(3x)/3+C | Linear inner function |
| ∫f(g(x))g'(x) dx | u = g(x) | ∫f(u) du | ∫2x·e^(x²) dx = e^(x²)+C | Product with derivative |
| ∫xⁿ·(xⁿ⁺¹)ᵐ dx | u = xⁿ⁺¹ | Power of power | ∫x·(x²+1)³ dx | Inner polynomial |
| ∫tan(x) dx | u = cos(x) | −ln|cos(x)|+C | ∫tan(x) dx | Trig quotient |
| Definite via sub | Change limits with u | New limits [g(a),g(b)] | ∫[0,1] 2x(1+x²)² dx | Update bounds too |
Interactive Tools
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Related Terms
Integration by Parts
Integration by parts is an integration technique derived from the product rule of differentiation, used to integrate products of two functions. The formula transforms an integral of a product into a simpler expression by strategically choosing which factor to differentiate (u) and which to integrate (dv). It is particularly effective for integrands involving products of polynomials with trigonometric, exponential, or logarithmic functions.
Chain Rule
The chain rule is a differentiation rule used to compute the derivative of a composite function, stating that the derivative of f(g(x)) equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. It is one of the most widely applied rules in calculus, essential whenever a function is "nested" inside another. The chain rule is critical in physics for related rates problems, in machine learning for backpropagation, and in multivariable calculus for total derivatives.
Integral
An integral is a mathematical object that represents the accumulation of quantities, such as areas under curves, total displacement, or accumulated change. There are two main types: the definite integral, which yields a numerical value representing the net area between a function and the x-axis over an interval, and the indefinite integral, which yields a family of antiderivative functions. Integration is the reverse process of differentiation and is one of the two fundamental operations of calculus.
The term "substitution" comes from Latin "substituere" meaning "to put in place of." The technique formalizes the inverse of the chain rule and was systematically developed alongside the formalization of integral calculus in the 17th–18th centuries.