4.4.2 U-Substitution

U-substitution is used for integrands in which one product term is the derivative of the other product term. That is, the integral has the form ∫v(u(x))⋅ⅆu(x)ⅆx ⅆx. When U-Substitution (see §3.4.1) is applied to a subexpression of the integrand, the integrand is transformed by replacing the subexpression with a placeholder variable, often u, and dividing the integrand by the derivative of the subexpression with respect to the original integrator. For an expression like ∫f(x)⋅.{g(x)} ⅆx, u-substitution produces

If u has been chosen appropriately, f_x and ⅆg_xⅆx will cancel, resulting in a simpler integral in terms of u. Moreover, the correspondence between the original subexpression and u is perpetuated in a decoration of the form u(x)→g_x.

Try this for ∫(2⋅x+5)⋅(x^2+5⋅x) ⅆx. Here, u(x)→x^2+5⋅x and v(x)→(2⋅x+5)⋅u(x). The derivative of the second product term in the integrand matches the first product term.

Select the second product term ∫(2⋅x+5)⋅(.{x^2+5⋅x.}) ⅆx and apply U-Substitution to produce

The strategy then is to remove all terms in x. In this case, the strategy is achieved by Simplify

The decoration on the integral is called the u-integrator. The name of the function in the u-integrator is generated so as not to be the same as any variable in either the integrand or the integrator. The right side is the original subject of u-substitution. When a decorated integral is integrated, the result also contains the decoration.

When a decorated integration is simplified, the right side of the decoration is substituted into the integrand:

You can prove the final result is correct by taking its derivative and simplifying it.

Decorated integrals use a special form of syntax. Normally, the integrator is coded as a single variable in the second operand to the integral, the one following ⅆ. A decorated integral codes the integrator using the form (u(x)→expression). A simpler form is also accepted: (u=expression). When the latter form is used, the arguments to the u-integrator are inferred from variables in the expression.

Note the use of parenthesis in the second operand of a decorated integral. If omitted, the expression ∫u ⅆu(x)=x^2+5*x would be parsed as an equation.

Try to simplify the following integrals:

1. ∫x⋅ⅇ^x^2 ⅆx
2. ∫cos x⋅ⅇ^sin x ⅆx
3. ∫cos (3⋅x+4) ⅆx
4. ∫1÷(1-2⋅x) ⅆx
5. ∫2⋅x⋅√(1+x^2) ⅆx
6. ∫4⋅x÷√(2⋅x^2+1) ⅆx

4.4.2.1 Integrator Substitution

The integral ∫sin (√x)÷√x ⅆx is solved using the third variation of u-substitution (§3.4.1), called integrator substitution. From the integral, extract the subexpression √x using selection and and use it to add u=√x to the algebra display. Select and Isolate x to produce the auxiliary expression u^2=x. Select the integrator ∫sin (√x)÷√x ⅆx and apply U-Substitution to produce ∫sin (√(u^2))÷√(u^2)⋅(2⋅u) ⅆu. This simplifies to ∫sin u⋅2 ⅆu and integrates to -2⋅cos u. Isolate u in the auxiliary expression and substitute it into the integration to yield -2⋅cos (√x).