Choosing a Substitution in a Mixed Integrand
An integrand may combine several kinds of expressions: a polynomial factor, a quotient, and a trigonometric or exponential function. The key question is not simply, “What expression can I call \(u\)?” It is, “Does the integrand contain the derivative of that expression, so the chain rule can reverse cleanly?”
A good substitution simplifies the integral while preserving the whole differential. In a mixed integrand, that may mean treating separate terms separately, rewriting a numerator, or deciding that a proposed substitution does not match.
A Reliable Selection Test
Start by looking for a composite expression: a function inside another function, such as \(x^2+1\) inside a cosine, or \(2x^3+1\) inside an exponential. Consider that inner expression as a candidate \(u\). Then differentiate it and compare its derivative with the remaining factors in the integrand.
Look for an expression raised to a power, inside a logarithm, exponential, trigonometric function, or denominator.
Calculate \(du=g'(x)\,dx\), including numerical factors.
Check whether the derivative is present, perhaps after factoring a constant or rewriting the integrand.
If the integrand is a sum, test its terms separately. If the match is incomplete, do not pretend the unmatched factor has disappeared.
The comparison is structural, not merely visual. For instance, if \(u=2x^3+1\), then \(du=6x^2\,dx\). A factor of \(x^2\) is present, but the factor of \(6\) must also be handled. Conversely, choosing an inner function whose derivative is absent usually makes the integral more complicated rather than simpler.
Why the Method Works
Substitution reverses the chain rule. If \(H\) is an antiderivative of \(F\), then \(H'(u)=F(u)\). Applying the chain rule to \(H(g(x))\) gives the justification.
Therefore, whenever the integrand has exactly the form \(F(g(x))g'(x)\), an antiderivative is \(H(g(x))+C\). If the integrand instead contains \(cF(g(x))g'(x)\), where \(c\) is a constant, its antiderivative is \(cH(g(x))+C\). This is why matching the differential—including its constant factor—is essential.
Worked Example: Separate Terms with Different Inner Functions
Worked Example: Separate Terms with Different Inner Functions
Evaluate \(\int \left(\frac{2x}{x^2+1}+6x^2e^{2x^3+1}\right)\,dx\).
Step 1: Treat the sum term by term. The first term contains the denominator \(x^2+1\), whose derivative is \(2x\). The second contains the exponent \(2x^3+1\), whose derivative is \(6x^2\). These are two different useful inner expressions.
Step 2: Substitute in the first term. Let \(u=x^2+1\). Then \(du=2x\,dx\), so
Since \(x^2+1\) is positive for every real \(x\), the absolute value could be omitted in the final logarithm.
Step 3: Substitute in the second term. Let \(v=2x^3+1\). Then \(dv=6x^2\,dx\), so
Step 4: Combine the results. By linearity, an antiderivative of the sum is
Check by differentiation:
This is the original integrand, so the antiderivative is verified.
The example illustrates an important choice: a sum need not have one inner function that controls every term. Linearity allows each term to be evaluated using the substitution that fits it. The constant of integration is written once for the combined antiderivative.
Worked Example: Use a Derivative Match in a Product
Worked Example: Use a Derivative Match in a Product
Evaluate \(\int (2x+3)\cos(x^2+3x+1)\,dx\).
Step 1: Identify the composite expression. The cosine contains \(x^2+3x+1\). Set \(u=x^2+3x+1\).
Step 2: Differentiate the inner expression.
The factor multiplying the cosine is exactly the derivative needed; no extra constant adjustment is required.
Step 3: Rewrite and integrate.
Check by differentiation:
This matches the original integrand, confirming the result.
Here the product is not a reason to reject substitution. The decisive fact is that the factor outside the cosine is precisely the derivative of its inner expression. A useful check is to place the proposed \(du\) beside the remaining factors and see whether every part of the original integrand has been accounted for.
Worked Example: Rewrite the Numerator Before Substituting
Worked Example: Rewrite the Numerator Before Substituting
Evaluate \(\int \frac{3x+5}{x^2+2x+5}\,dx\).
Step 1: Compare numerator and denominator derivative. Let \(D=x^2+2x+5\). Then \(D'=2x+2\), which is not the numerator \(3x+5\). A direct substitution for the whole numerator therefore does not match.
Step 2: Split the numerator into a matching part and a remainder.
Thus the integral becomes
Step 3: Substitute in the matching part. In the first integral, use \(u=x^2+2x+5\), so \(du=(2x+2)\,dx\). It becomes \(\frac{3}{2}\int du/u=\frac{3}{2}\ln|u|\). The denominator is positive because
so the absolute value is unnecessary.
Step 4: Evaluate the remaining standard form. Completing the square gives \(x^2+2x+5=(x+1)^2+4\). Therefore
because differentiating the arctangent expression gives \(2/((x+1)^2+4)\).
Step 5: Combine and check. An antiderivative is
Differentiating its first term gives \(\frac{3x+3}{x^2+2x+5}\). Differentiating its second gives \(\frac{2}{(x+1)^2+4}=\frac{2}{x^2+2x+5}\). Their sum is
the original integrand.
This example shows that a substitution can handle only part of an integrand. Rewriting the numerator exposes a derivative match, while the remaining term is evaluated separately. It would be incorrect to claim that substituting \(u=x^2+2x+5\) turns the entire original integral into a constant multiple of \(\int du/u\), because the leftover \(2\) in the numerator does not become part of \(du\).
Common Mistake: Losing the Derivative Factor
A frequent error is to recognize a composite expression but omit its derivative when applying the chain rule in reverse. For example, consider \(\int 2x\cos(x^2+1)\,dx\). The substitution \(u=x^2+1\) gives \(du=2x\,dx\), and the integral is \(\int\cos u\,du=\sin u+C\).
A wrong calculation would claim that \(\int\cos(x^2+1)\,dx=\sin(x^2+1)+C\). Differentiating the claimed answer gives \(2x\cos(x^2+1)\), not \(\cos(x^2+1)\). The correction is to compare the derivative of the proposed antiderivative with the actual integrand: the formula \(\sin(x^2+1)\) works for the integral containing the factor \(2x\), not for the integral without it.
When a Candidate Does Not Match
If the derivative of a candidate inner function is absent, first check whether the integrand can be split or rewritten without changing its value. A constant factor may be pulled out, and a numerator may sometimes be expressed as a multiple of the denominator’s derivative plus a remainder. But do not insert a missing factor or treat an unmatched term as if it were part of \(du\).
Not every integral becomes simpler by substitution. If no candidate produces a useful match, the correct conclusion is that this substitution is not justified by the integrand as written. Recognizing that limit is part of choosing a method carefully, not a failure to complete the calculation.
Check Your Understanding
For each question, identify the useful inner expression, if there is one, and explain how you would check the result.
- For \(\int 4x(x^2+3)^5\,dx\), what substitution matches the differential, and what constant factor, if any, remains?
- In \(\int \left(2x e^{x^2}+\frac{1}{x+1}\right)\,dx\), why is it reasonable to treat the two terms separately?
- For \(\int (3x^2)\sin(x^3+2)\,dx\), what is \(du\) if \(u=x^3+2\), and what does the integral become?
- If a proposed substitution accounts for the denominator’s derivative but leaves an extra numerator term, what should you do rather than ignore that term?
- What calculation verifies that a proposed indefinite integral is correct?
