MathematicsCalculusA-Level

CBSEGCE A-LevelWJECAbiturAPAQABaccalauréat GénéralBachillerato

Integration by Substitution

Reverse chain rule for integration.

Understand the formulaSee the free derivationOpen the full walkthrough

\int f (g (x)) g^{'} (x) d x = \int f (u) d u

Open Full Walkthrough Try Calculator

This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.

Core idea

Overview

Integration by substitution is a formal method in calculus used to simplify the integration of composite functions by changing the variable of integration. It serves as the integral equivalent of the chain rule, transforming a complex integrand into a simpler form where the antiderivative is more easily recognized. By identifying a function and its derivative within the integrand, the variable is shifted to u, streamlining the calculation process.

When to use: Apply this method when the integrand contains a function and its derivative, typically in the form of a composite function. It is particularly useful when dealing with powers of polynomials, trigonometric identities, or exponential terms where the exponent is non-linear.

Why it matters: This technique is essential for solving complex differential equations found in physics, such as those governing planetary motion or electromagnetism. It allows scientists to solve integrals that are otherwise impossible to evaluate, providing a bridge between symbolic representations and numerical solutions.

Symbols

Variables

k = Coefficient k, n = Power n, a = Lower limit a, b = Upper limit b, I = Integral result

k

Coefficient k

V a r iab l e

n

Power n

V a r iab l e

a

Lower limit a

V a r iab l e

b

Upper limit b

V a r iab l e

I

Integral result

V a r iab l e

Walkthrough

Derivation

Understanding Integration by Substitution

Substitution reverses the chain rule by changing variables to turn a complicated integral into a simpler one.

The integrand contains a composite function and its derivative (up to a constant multiple).

Identify a Substitution:

Choose u as an inner function whose derivative also appears in the integrand.

\int f (g (x)) g^{'} (x) d x, let u = g (x)

Differentiate to Relate du and dx:

This allows you to replace $g^{'} (x) d x$ with du.

\frac{d u}{d x} = g^{'} (x) ⟹ d u = g^{'} (x) d x

Rewrite the Integral in u:

After substitution, integrate with respect to u, then convert back to x if needed.

\int f (u) d u

Result

\int f (u) d u

Source: Standard curriculum — A-Level Pure Mathematics (Integration)

Visual intuition

Graph

Graph unavailable for this formula.

The graph of u = g(x) is typically a linear function when applying the simplest form of substitution, though it can take the shape of any polynomial or transcendental function depending on the chosen substitution. The curve represents a change of variable that maps the complex integrand into a simpler form, where the slope at any point corresponds to the derivative du/dx. This transformation effectively re-scales the horizontal axis to simplify the process of finding the antiderivative.

Graph type: linear

Why it behaves this way

Intuition

Imagine stretching or compressing the x-axis to transform a complex area under a curve into a simpler, more recognizable shape whose area is easier to calculate.

A new variable representing the inner function g(x)

Renaming a complex part of the integrand to a simpler variable to make the expression easier to work with

The differential of the new variable u, which replaces g'(x) dx

The 'scaling factor' that accounts for the change in the integration variable, derived from the relationship du/dx = g'(x)

g(x)

The inner function within the composite function f(g(x))

The specific part of the integrand chosen to be replaced by the new variable u, simplifying the 'core' of the expression

g'(x)

The derivative of the inner function g(x)

The necessary factor in the integrand that allows for the substitution du = g'(x) dx, acting as a 'matching piece' for the differential transformation

Free study cues

Insight

Canonical usage

This method ensures that the units of the integrated expression remain consistent across the variable transformation, maintaining dimensional homogeneity.

Common confusion

A common mistake is failing to correctly transform the differential (dx to du) or to ensure that the units of the new integrand f(u)du are consistent with the original f(g(x))g'(x)dx.

Dimension note

While the equation itself describes a mathematical transformation, the variables and functions involved can carry physical units. The core principle is that the dimensions of the integrand on both sides of the equation

One free problem

Practice Problem

Evaluate the definite integral of 2x(x² + 1)² dx from x = 0 to x = 1.

lower_limit0

upper_limit1

Solve for: $I$

Hint: Substitute u = x² + 1.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Transforming coordinates.

Study smarter

Tips

Identify the 'inner' function whose derivative exists elsewhere in the integrand.
Always compute the differential du and solve for dx if necessary.
Remember to transform the upper and lower limits of integration when working with definite integrals.
Simplify the resulting expression in terms of u before performing the final integration.

Avoid these traps

Common Mistakes

Not replacing dx with du terms.
Leaving x's in u integral.

Common questions

Frequently Asked Questions

Substitution reverses the chain rule by changing variables to turn a complicated integral into a simpler one.

Apply this method when the integrand contains a function and its derivative, typically in the form of a composite function. It is particularly useful when dealing with powers of polynomials, trigonometric identities, or exponential terms where the exponent is non-linear.

This technique is essential for solving complex differential equations found in physics, such as those governing planetary motion or electromagnetism. It allows scientists to solve integrals that are otherwise impossible to evaluate, providing a bridge between symbolic representations and numerical solutions.

Not replacing dx with du terms. Leaving x's in u integral.