Matching a Power to the Derivative
Substitution is useful when part of an integrand is the derivative of an expression appearing inside another function. For an inner expression \(x^n\), the derivative is \(n x^{n-1}\). That means the power \(x^{n-1}\) is the exact power that makes the remaining factor proportional to the differential \(du\).
This is a specific pattern within the substitution method. It lets us integrate a composite expression \(F(x^n)\) when the integrand also contains the matching power of \(x\). If the power differs, substitution may leave an extra factor involving \(x\), and the match is no longer direct.
The Direct Substitution Formula
Suppose \(F\) has an antiderivative \(H\) on the relevant range, so \(H'(u)=F(u)\). With \(u=x^n\), the chain rule gives
Dividing by \(n\) shows why the matching power gives a direct antiderivative. The formula holds on any interval where the functions involved are defined.
The factor \(1/n\) is essential: the derivative of \(x^n\) is not just \(x^{n-1}\), but \(n x^{n-1}\). A constant multiple in the integrand can be carried through the same calculation.
Worked Example: A Composite Power With the Matching Factor
Evaluate \(\int x^4(x^5+2)^3\,dx\). The inner expression is \(x^5+2\), whose derivative is \(5x^4\). The integrand contains \(x^4\), which matches the derivative power.
Set \(u=x^5+2\). Then \(du=5x^4\,dx\), so \(x^4\,dx=\frac{1}{5}du\). Therefore
Check by differentiation:
The derivative is the original integrand, so the antiderivative is correct.
How to Test the Power
For an integrand of the form \(x^mF(x^n)\), first differentiate the inner expression \(x^n\). Its derivative contributes \(n x^{n-1}\). Compare the exponent \(m\) with \(n-1\), not merely with \(n\) or with the exponent inside \(F\).
Read the composite factor as \(F(x^n)\) and note the positive integer \(n\).
The derivative of \(x^n\) is \(n x^{n-1}\).
If \(m=n-1\), the factor \(x^m\,dx\) is exactly \(\frac{1}{n}du\). If not, check whether a residual power can be written in terms of \(u\), or whether another method is needed.
Worked Example: Including a Constant Multiple
Evaluate \(\int 6x^5(x^6+1)^2\,dx\). Here the inner expression is \(x^6+1\), and its derivative is \(6x^5\). The integrand contains that full derivative factor.
Set \(u=x^6+1\), so \(du=6x^5\,dx\). Then
Check by differentiating the result:
This recovers the original integrand exactly.
What If the Power Does Not Match?
If \(m\ne n-1\), then substituting \(u=x^n\) does not turn all of \(x^m\,dx\) into a constant multiple of \(du\). Indeed, from \(du=n x^{n-1}\,dx\), we obtain
The remaining factor \(x^{m-n+1}\) is the issue: it must not be silently discarded. In some cases it can itself be rewritten as a power of \(u\). For example, if \(m-n+1=nk\) for an integer \(k\), then \(x^{m-n+1}=x^{nk}=u^k\). This produces an integral in \(u\), but it is no longer just a constant multiple of \(F(u)\); the factor \(u^k\) remains.
For negative integer \(k\), this rewrite requires \(x\ne 0\), so that \(u\ne 0\). The original integrand and interval must also be within their domains. When the residual exponent is not a suitable multiple of \(n\), substitution may leave an \(x\)-factor that cannot be expressed this way.
Worked Example: A Residual Power That Can Be Rewritten
Evaluate \(\int x^5(1+x^3)^2\,dx\). The inner power is \(x^3\), so \(n=3\) and \(du=3x^2\,dx\). The outside exponent is \(m=5\), not \(n-1=2\), so this is not a direct match. But \(x^5\,dx=x^3(x^2\,dx)\), and \(x^3=u\) after setting \(u=x^3\).
Thus \(x^2\,dx=\frac{1}{3}du\), and the integral becomes
Integrating each term gives
Substitute \(u=x^3\):
Check by differentiation:
The residual factor became \(u\), which left a polynomial in \(u\) that could be integrated.
When the Residual Factor Does Not Help
A substitution is not automatically useful just because an inner power is visible. If the unmatched factor cannot be expressed in terms of \(u\), the substitution has not reduced the problem to an integral entirely in \(u\). One should then consider a different method, or use another form of the integrand if available.
Worked Example: An Unmatched Power
Consider \(\int x^4(1+x^3)^2\,dx\). The inner expression \(x^3\) has derivative \(3x^2\), but the outside power is \(x^4\), so \(m=4\ne 2=n-1\). The leftover factor from the substitution is \(x^{4-3+1}=x^2\), which is not a power \(x^{3k}\) with integer \(k\). The substitution does not directly eliminate \(x\).
In this example, expanding the polynomial gives a straightforward alternative:
Integrating term by term yields
Check by differentiation:
The integral is manageable, but the direct \(u=x^3\) match was absent. Recognizing that distinction prevents an incomplete substitution.
Common Mistake: Dropping the Chain Factor
A frequent error is to match \(x^{n-1}\) but forget the coefficient \(n\) in \(du=n x^{n-1}\,dx\). For instance, the incorrect calculation
omits the factor needed to compensate for differentiating \(x^5\). Differentiating this incorrect answer exposes the problem:
which is five times the integrand. The correction is to divide by \(5\), giving \(\frac{(x^5+2)^4}{20}+C\). Differentiating that expression returns the integrand exactly.
Why the Match Matters
The exponent test is a quick way to decide whether a composite expression is ready for a direct substitution. For \(F(x^n)\), the matching outside power is always \(x^{n-1}\); the coefficient \(n\) is handled as a constant. When \(m\) differs, the formula \(x^m\,dx=\frac{1}{n}x^{m-n+1}\,du\) makes clear precisely what remains to be addressed.
These observations do not guarantee that every integral can be evaluated in elementary functions. They identify the structure of the substitution and prevent two common errors: overlooking the chain factor and treating a leftover \(x\)-power as if it had disappeared. The final check is always to differentiate the proposed antiderivative and compare it with the original integrand.
Check Your Understanding
Answer these questions by identifying the inner power and comparing exponents.
- For \(\int x^6F(x^7)\,dx\), does \(u=x^7\) give a direct match? What constant factor relates \(x^6\,dx\) to \(du\)?
- For \(\int x^3F(x^5)\,dx\), what power of \(x\) remains after expressing \(x^3\,dx\) using \(du\)?
- If \(u=x^4\), which power \(x^m\) makes \(x^m\,dx\) directly proportional to \(du\)?
- For \(\int x^8(1+x^3)^2\,dx\), can the residual factor from \(u=x^3\) be written as an integer power of \(u\)? Explain using the residual exponent.
- Why should a proposed answer to a substitution problem be checked by differentiation?
