Chapter 15 — Key Takeaways

The Two Major Techniques

Chapter 15 gives you the two most important integration techniques, each a differentiation rule read backwards:

• $u$-substitution reverses the chain rule. Use it for a composite function $f(g(x))$ multiplied by (a constant times) the derivative $g'(x)$ of the inner function. (Sections 15.1–15.3)
• Integration by parts reverses the product rule, $\displaystyle\int u\,dv = uv - \int v\,du$. Use it for a product of two different types of functions. (Sections 15.4–15.6)

Everything else in the chapter is refinement: changing limits, repeating the methods, the rotating trick, and combining the two.

$u$-Substitution — Recognition and Execution

The fingerprint to hunt for: an inner function and its derivative both present, up to a constant factor — the structure $f'(g(x))\cdot g'(x)$. (Section 15.1)

The seven-step procedure:

1. Recognize the composite-with-inner-derivative pattern.
2. Choose $u = g(x)$ (the inner function).
3. Compute $du = g'(x)\,dx$.
4. Substitute so the integral is entirely in $u$ — no $x$ may remain.
5. Integrate the simpler $u$-integral.
6. Back-substitute $u = g(x)$.
7. Verify by differentiating.

Patterns worth memorizing as reflexes:

• $\displaystyle\int \frac{g'(x)}{g(x)}\,dx = \ln|g(x)| + C$ — derivative over function gives a logarithm. (Section 15.2)
• Linear inner function $u = ax+b$: the answer is always $\tfrac{1}{a}$ times "what you'd get if the inner were just $x$." So $\int e^{ax}\,dx = \tfrac1a e^{ax}+C$, $\int\sin(ax)\,dx = -\tfrac1a\cos(ax)+C$. (Section 15.2)
• A power of $\sin$ times one $\cos$ (or vice versa): peel off the lone factor as $du$. (Section 15.2)

Handling a leftover $x$: if an $x$ refuses to leave after substituting, solve the substitution backwards (e.g. $u = x+1 \Rightarrow x = u-1$, or $u = x^2+1 \Rightarrow x^2 = u-1$) and express it in $u$. (Sections 15.1–15.2)

Definite Integrals by Substitution

Two valid approaches (Section 15.3):

• Approach 1 — antiderivative first. Find the antiderivative, back-substitute fully to $x$, then apply the original limits.
• Approach 2 — change the limits. Replace the variable and the limits together: $x=a$ becomes $u=g(a)$, $x=b$ becomes $u=g(b)$. Never return to $x$. Usually cleaner.

The cardinal rule: the limits must always match the variable in the bracket. Never evaluate a $u$-antiderivative at the original $x$-limits.

Shortcuts before computing: check for parity — an odd integrand over a symmetric interval $[-a,a]$ integrates to $0$. And glance at whether the transformed integrand is well-behaved; a substitution can reveal a hidden improper integral (handled in Chapter 17), as in $\int_0^{\pi/4}\tan(2x)\,dx$.

Integration by Parts — LIATE and Beyond

The formula: $\displaystyle\int u\,dv = uv - \int v\,du$. It trades one integral for another; the art is making the new one easier. (Section 15.4)

The LIATE heuristic — let $u$ be the first type that appears, in order of preference:

Whatever remains is $dv$. The logic: $u$ gets differentiated, so pick the factor that becomes simpler when differentiated (logs and inverse trig simplify; polynomials drop their degree). Trig and exponentials merely cycle, so they make good $dv$. (Section 15.4)

Standard results to memorize (Sections 15.5, 15.11):

• $\displaystyle\int \ln x\,dx = x\ln x - x + C$ (the "invisible $1$" trick: write $\ln x = \ln x \cdot 1$).
• $\displaystyle\int \arctan x\,dx = x\arctan x - \tfrac12\ln(1+x^2) + C$.
• $\displaystyle\int x e^x\,dx = (x-1)e^x + C$.

Repeated parts and the tabular method: $\int x^n e^x\,dx$ or $\int x^n \sin x\,dx$ needs $n$ passes. The tabular method organizes them: a sign column ($+,-,+,\dots$), repeated derivatives of the polynomial (down to $0$), and repeated integrals of the other factor; multiply along the diagonals. Always put the polynomial in the derivative column so the table terminates. (Section 15.5)

The rotating trick: when two passes regenerate the original integral $I$ (e.g. $\int e^x\sin x\,dx$, $\int\sec^3 x\,dx$), treat $I$ as an unknown and solve algebraically. You must keep the same choice of $u$ on the second pass — switching undoes the first pass and gives the useless $I = I$. (Section 15.6)

When to Use Which — Strategy

Run this checklist, roughly in order of frequency (Section 15.7):

1. Already in the table? Write it down.
2. Composite $f(g(x))$ with $g'(x)$ present? → $u$-substitution.
3. Product of two different types? → integration by parts (LIATE).
4. Powers/products of trig functions → Chapter 16 (trigonometric integrals).
5. Radicals $\sqrt{a^2\pm x^2}$, $\sqrt{x^2-a^2}$ → Chapter 16 (trigonometric substitution).
6. Rational function with no obvious substitution → Chapter 16 (partial fractions).
7. No elementary antiderivative ($\int e^{-x^2}\,dx$) → Chapter 16 (numerical methods).

Spend the first ten seconds classifying, not computing. The two techniques are partners, not rivals: parts often hands you a substitution as its second act (Section 15.5), and hard integrals frequently need both in sequence (Section 15.8).

Common Errors to Avoid

• Fixing a missing factor with a variable. You may adjust by a constant ($\int x\cos x^2\,dx \to \tfrac12\int\cos u\,du$), never by a variable ($\tfrac{1}{2x}$ cannot leave the integral). (Section 15.2)
• Mixing $u$-antiderivatives with $x$-limits in a definite integral. Change the limits the moment you change the variable. (Section 15.3)
• Choosing $u$ against LIATE, so the new integral is worse (the power of $x$ goes up). (Section 15.4)
• Switching direction in the rotating trick, collapsing to $I = I$. (Section 15.6)
• Sign slips in repeated parts — the exact problem the tabular method prevents. (Section 15.5)
• Forgetting $+C$ (indefinite) or the absolute-value bars in $\ln|g(x)|$.

Connections

• Backwards to Chapter 14: these techniques find the antiderivative that FTC requires; the AUC, work, and expected-value integrals are all the net-change reading of FTC.
• Backwards to Chapter 7: $u$-substitution is the chain rule reversed; integration by parts is the product rule reversed.
• Forward to Chapter 16: the four specialized methods (trig integrals, trig substitution, partial fractions, numerical) for integrals these two techniques cannot close.
• Forward to Chapter 17: improper integrals — several of this chapter's applications (AUC, expected value, Laplace transforms) quietly ran a limit to infinity.
• Forward to Chapters 18–19: applications to area, volume, arc length, work (where $\int\sec^3 x\,dx$ resurfaces), and solving differential equations.
• Forward to Chapter 33: $u$-substitution is the one-dimensional ancestor of the multivariable change of variables and the Jacobian.

Bottom line. Master the two reverse-rules and you can integrate the large majority of functions in scientific work. Where they fail, they fail informatively — telling you exactly which Chapter 16 method to reach for next.

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