AP Calculus AB/BC · Unit 6 · Topic 6.9
Integrating Using Substitution
Reverse the chain rule by replacing an inner expression and its differential.
1. Topic Focus
Interpret definite integrals as accumulated change, connect sums to integrals, apply both Fundamental Theorems, and select antiderivative techniques.
This topic: Reverse the chain rule by replacing an inner expression and its differential.
2. Key Relationship
Read every symbol with its domain, direction, units, and hypotheses before applying the relationship.
3. Visual Connection
4. Worked Example
For ∫2x cos(x²)dx, let u=x² to obtain sin(x²)+C.
Write the governing relationship first, carry out the algebra cleanly, and finish with a sentence that answers the mathematical question.
5. Concept Development
Substitution reverses the chain rule
The derivative of \(F(g(x))\) is \(F'(g(x))g'(x)\). Integration by substitution recognizes that chain-rule structure and replaces the inner expression with a simpler variable.
Choose the inner expression that simplifies the integral
Useful candidates commonly appear inside a power, radical, exponential, logarithm, denominator, or trigonometric function. Differentiate the candidate and compare it with the remaining factors.
The differential replaces a complete factor
After setting \(u=g(x)\), write \(du=g'(x)dx\). Treat \(g'(x)dx\) as one package and determine exactly what constant multiple is present in the original integrand.
Constant-factor adjustments are allowed
If \(du=6x,dx\) but the integral contains \(x,dx\), then \(x,dx=\frac16du\). Constants may move outside the integral; variable factors may not.
Every part of the integrand must use one variable
A completed substitution contains only \(u\) and \(du\). If an \(x\) remains, rewrite it using the substitution or choose a more effective substitution.
Indefinite integrals return to the original variable
Integrate in \(u\), substitute \(u=g(x)\) back into the answer, and include \(+C\). The final antiderivative should normally be expressed in the variable from the original problem.
Definite integrals require corresponding new bounds
Evaluate the substitution at each original endpoint before integrating in \(u\).
Do not mix variables and bounds
A \(u\)-integral must have \(u\)-bounds. If the antiderivative is converted back to \(x\), use the original \(x\)-bounds instead. Either route is valid, but combining them is not.
A decreasing substitution may reverse the new bounds
If \(g(a)>g(b)\), the transformed lower bound is larger than the upper bound. Keep that order and allow orientation to handle the sign rather than silently rearranging the bounds.
Power patterns are common substitution targets
The logarithmic pattern uses a function over itself
The absolute value is required unless the domain guarantees that \(g(x)>0\).
Exponential patterns preserve the inner expression
For \(a^{g(x)}\), divide by \(\ln a\) after substitution.
Trigonometric substitutions reverse chain-rule derivatives
For example, \(\int\cos(g(x))g'(x)dx=\sin(g(x))+C\), while a sine pattern introduces a negative cosine. Signs should be checked by differentiation.
Algebra may be needed after choosing u
Sometimes the substitution lets you solve for a remaining factor. If \(u=x+1\), then \(x=u-1\); if \(u=x^2+1\), then \(x^2=u-1\). Rewrite before integrating.
Substitution is not always the right method
The presence of a composite function alone is insufficient. A matching derivative factor must be available or obtainable by a constant adjustment or valid algebraic rewrite.
Differentiate to verify an indefinite result
Applying the chain rule to the final expression should reproduce the original integrand exactly. This detects missing constants, incorrect signs, and incomplete back-substitution.
A reliable substitution workflow
Select \(u\), compute \(du\), adjust constants, rewrite everything in \(u\), transform bounds when present, integrate, return to \(x\) only when needed, and verify.
Common errors
Frequent errors include choosing an inner function without its derivative, dropping a coefficient, leaving mixed variables, using original bounds on a \(u\)-integral, adding \(+C\) to a definite value, and changing reversed bounds without changing sign.
6. Detailed Worked Example and Error Check
Example 1: Reverse a polynomial chain rule.
where \(u=3x^2+5\) and \(du=6x\,dx\).
Example 2: Adjust a missing constant. Let \(u=x^3+4\), so \(du=3x^2dx\). Then
Example 3: Recognize the logarithmic pattern.
The numerator is exactly the derivative of the denominator, which is always positive.
Example 4: Track a trigonometric sign. With \(u=\cos x\) and \(du=-\sin xdx\),
Example 5: Change the bounds of a definite integral. For \(u=x^2+1\), the bounds \(x=0,2\) become \(u=1,5\):
Example 6: Preserve decreasing-bound orientation. Let \(u=\cos x\), so \(du=-\sin xdx\). The bounds \(0\) and \(\pi/2\) become \(1\) and \(0\):
Example 7: Rewrite a remaining variable. Set \(u=x+1\), so \(x=u-1\) and \(dx=du\). Then
7. AP Reasoning Routine
Identify the accumulating quantity and units, preserve bounds, choose a valid integration technique, and check answers by differentiation.
- Identify the representation and requested quantity.
- State the rule or theorem and verify its conditions.
- Keep exact values until the final requested approximation.
- Interpret sign, units, interval, and context.
Use substitution and show the choice of \(u\), the differential, and any transformed bounds.
(a) Evaluate \(\int8x(4x^2-3)^6dx\).
(b) Evaluate \(\int\frac{x^2}{x^3+2}dx\).
(c) Evaluate \(\int e^{5x-1}dx\).
(d) Evaluate \(\int\cos(3x)dx\).
(e) Evaluate \(\int\frac{\cos x}{2+\sin x}dx\).
(f) Evaluate \(\int_1^3\frac{2x}{x^2+4}dx\) using transformed bounds.
(g) Evaluate \(\int_0^1 3x^2\sqrt{x^3+1}\,dx\).
(h) Evaluate \(\int_0^{\pi/2}\cos x(1+\sin x)^2dx\).
(i) A student writes \(u=x^2+1\) and then \(\frac12\int_0^2u^3du\). Explain the notation error for the original integral \(\int_0^2x(x^2+1)^3dx\).
(j) Evaluate \(\int x^3\sqrt{x^2+1}\,dx\) by rewriting the remaining \(x^2\) after substitution.
Check the solution
(a) Let \(u=4x^2-3\), so \(du=8x\,dx\). The result is \((4x^2-3)^7/7+C\).
(b) Let \(u=x^3+2\), so \(du=3x^2dx\). The result is \(\frac13\ln|x^3+2|+C\).
(c) Let \(u=5x-1\), so \(dx=du/5\). The result is \(\frac15e^{5x-1}+C\).
(d) With \(u=3x\), the result is \(\frac13\sin(3x)+C\).
(e) Let \(u=2+\sin x\), so \(du=\cos xdx\). The result is \(\ln|2+\sin x|+C\).
(f) Let \(u=x^2+4\); the bounds become 5 and 13. Thus \(\int_5^{13}du/u=\ln(13/5)\).
(g) Let \(u=x^3+1\); the bounds become 1 and 2. The value is \(\int_1^2u^{1/2}du=\frac23(2\sqrt2-1)\).
(h) Let \(u=1+\sin x\); the bounds become 1 and 2. The value is \(\int_1^2u^2du=7/3\).
(i) After changing to \(u\), the limits must also change from \(x=0,2\) to \(u=1,5\). The correct transformed integral is \(\frac12\int_1^5u^3du\).
(j) Let \(u=x^2+1\), so \(x^2=u-1\) and \(x\,dx=du/2\). Then \(\frac12\int(u-1)u^{1/2}du=\frac15u^{5/2}-\frac13u^{3/2}+C\), giving \(\frac15(x^2+1)^{5/2}-\frac13(x^2+1)^{3/2}+C\).