Nested Compositions In Indefinite Integrals

Why Nested Expressions Need Careful Substitution

An expression such as \((5x^2-1)^6\) is built in layers: first \(x\) is squared and combined with constants, then the resulting expression is raised to a power. The chain rule differentiates these layers from the outside inward. Substitution reverses that process: we look for an inner expression whose derivative is also present in the integrand.

The key is not to choose an expression merely because it appears inside another one. A useful choice makes the rest of the integrand expressible in terms of the new variable, with its differential accounted for. In a nested expression, the argument of the outer function is often the useful choice. We will apply this idea to powers, exponentials, and trigonometric functions.

The Rule Behind the Method

Suppose \(F\) is continuous on an interval and \(H\) is an antiderivative of \(F\) there, so \(H'(u)=F(u)\). Let \(g\) be differentiable, with its values in that interval. The chain rule gives

Therefore, whenever an integrand has the form \(F(g(x))g'(x)\), an antiderivative is \(H(g(x))\). In substitution notation, we set \(u=g(x)\) and \(du=g'(x)\,dx\), so the integral becomes \(\int F(u)\,du\). We then find an antiderivative in \(u\) and replace \(u\) by \(g(x)\). The arbitrary constant \(C\) is included because all antiderivatives of the same function on an interval differ by a constant.

This result explains why substitution works: it is precisely the chain rule run backward. The derivative factor \(g'(x)\) is essential. A constant multiple of it is also enough, provided that constant is carried through the calculation.

A Practical Way to Choose \(u\)

In a composite such as \(F(g(x))\), begin by considering \(u=g(x)\), the argument being fed into the outer function \(F\). Then calculate \(du\) and compare it with the other factors in the integrand. A successful choice has two features:

• The outer function becomes a familiar expression in \(u\), such as \(u^n\), \(e^u\), or \(\sin u\).
• The remaining factors can be replaced by a constant multiple of \(du\), leaving an integral entirely in \(u\).

The derivative factor may be written in a different-looking form, so compare it algebraically rather than relying only on visual matching. If the factor is a constant multiple of \(g'(x)\), that constant remains as a multiplier or divisor in the transformed integral.

Worked Example: A Nested Power

Worked Example: An Exponential of a Nested Expression

Worked Example: A Trigonometric Function of a Trigonometric Expression

Why the Differential Factor Matters

The substitution does not replace an expression in isolation. It replaces both the chosen expression and its derivative factor. For example, if \(u=g(x)\), then \(du=g'(x)\,dx\); it is not generally valid to replace \(g(x)\) by \(u\) while leaving every other factor untouched. The transformed integral must contain only \(u\) and \(du\).

A common error is to omit the constant multiplier in the differential. In the nested-sines example, treating \(\cos(3x)\,dx\) as \(du\) would give the incorrect result \(-\cos(\sin(3x))+C\). Differentiating that proposed answer exposes the error:

which is three times the required integrand. Since \(du=3\cos(3x)\,dx\), the correct replacement is \(\cos(3x)\,dx=\frac{1}{3}du\). Including that factor produces \(-\frac{1}{3}\cos(\sin(3x))+C\), whose derivative was checked above.

When a Choice Is Not Yet Useful

A substitution is useful only if it accounts for the whole integrand. Suppose one sets \(u=3x^2+2\) in the exponential example. The exponential would become \(e^{u^3}\), and the remaining power would become \(u^2\). That substitution does not turn the integral into a basic exponential integral in \(u\). The choice \(u=(3x^2+2)^3\), by contrast, makes the exponential \(e^u\) and its derivative supplies the other factors up to a constant.

This comparison illustrates a practical test: after proposing \(u\), write out \(du\) and rewrite every factor in the integral. If some part remains in terms of \(x\), or if the resulting integral is not simpler, the choice has not accomplished the intended reversal of the chain rule. Not every nested expression has a direct substitution of this form.