3.4.1 U-Substitution

U-substitution is a calculus technique for integration. It performs three different transformations, depending on the context of the subject.

1. If the subject is a subexpression of an integrand, U-substitution uses the subexpression to synthesize a function.
2. If the subject is the decoration on an integral that is the result of a previous u-substitution, an inverse transformation is performed, recovering the original integral.
3. If the subject is an integrator, other expressions on the display are harvested for a function definition to replace subexpressions in the integrand. This form of u-substitution is called integrator substitution.

The three transformations are described here. Section §4.4.2 contains more detailed explanations along with examples.

3.4.1.1 Synthesized U-substitution

The usual calculus technique of u-substitution is employed by selecting a subexpression of the integrand and applying U-Substitution . The subexpression becomes the right side of a synthesized replacement function, depicted here by u(x)→expr. The integrand is rewritten by replacing the integrator ⅆx:1 by 1÷ⅆexprⅆx⋅ⅆu:1, which can be seen to be the isolation of ⅆx:1 in the identity ⅆuⅆx=ⅆexprⅆx. The synthesized function is added as a decoration to the integral.

The next step is to simplify the derivative introduced by 1÷ⅆexprⅆx⋅ⅆu:1. Because this next step is inevitable, it is performed automatically by the u-substitution transformation.

For example, applied to a subexpression of the integrand in ∫.{sin x}⋅cos x ⅆx, U-Substitution transforms the integral into ∫u⋅cos x÷cos x ⅆ(u(x)→sin x). Simplification yields ∫u ⅆ(u(x)→sin x), which is more easily integrated.

3.4.1.2 Inverse U-substitution

Sometimes it is necessary to work backwards from an expression that contains an integral with a decoration. Applied to a decoration, U-substitution performs an inverse operation, replacing all occurrences of the u expression with the right side of the definition. The integrator is augmented by an isolation of ⅆu:1 in the identity ⅆuⅆx=ⅆexprⅆx.

For example, with the decoration in ∫u ⅆ.{u(x)→sin x} selected, U-substitution produces

∫sin x⋅cos x ⅆx

3.4.1.3 Integrator Substitution

Integrator substitution is used when the substituted expression is not a subexpression of the integrand. Applied when the integrator is the subject, U-substitution harvests a candidate equation from the other expressions in the workspace. A successful candidate is supplied by any equation with the integrator on one side and a substitution expression involving a single variable on the other. The integrand is transformed by replacing all occurrences of the integrator with the substitution expression and replacing the integrator by the left side. Finally, the integrand is augmented with the derivative of the function.

For example, consider u-substitution on the integrator of ∫x^2⋅√(4-x^2) ⅆx in the presence of a candidate equation x=2⋅sin t. All occurrences of x will be replaced by sin t, as will the integrator. The augment is x=2⋅sin t. The result is

If the integral has bounds, the single variable in the candidate equation is isolated to obtain a mapping expression in terms of the original integrator. The bounds are transformed by applying the mapping expression.

To see how this works, consider the simple integral ∫-1, ℼ, x ⅆx and the equation √y=x. Integrator substitution produces ∫-1^2, ℼ^2, √y⋅ⅆ√yⅆy ⅆy. Simplification produces ∫-1^2, ℼ^2, 1/2 ⅆy. Simplification and evaluation produces the final result (ℼ^2-1)÷2. The original integral simplified directly produces the same result.