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\( \int \)

Selecting Techniques for
Antidifferentiation

Learn to identify which integration technique to apply: u-substitution, integration by parts, partial fractions, and more.

AP Calculus BC

What Is Antidifferentiation?

Antidifferentiation (also called indefinite integration) is the process of finding a function \( F(x) \) whose derivative equals a given function \( f(x) \). In symbols:

Unlike differentiation, which follows algorithmic rules, antidifferentiation often requires you to choose the right technique based on the structure of the integrand.

Why Technique Selection Matters

The AP Calculus BC exam expects you to quickly recognize which method fits a given integral. Choosing the wrong technique wastes time or leads to dead ends.

Speed
The right technique can reduce a complex integral to one or two lines of work.
Accuracy
Misidentifying the form leads to algebraic errors and incorrect antiderivatives.
Flexibility
Some integrals can be solved by multiple methods: knowing all of them gives you options.

Your Integration Toolbox

AP Calculus BC covers these core antidifferentiation techniques. Each works best on a specific structural pattern in the integrand.

\( u \)-Substitution
Reverse chain rule: integrand contains \( g(x) \) and its derivative \( g'(x) \).
By Parts
Product of two "different-type" functions: e.g., \( x\,e^x \) or \( x\,\ln x \).
Partial Fractions
Rational function with factorable denominator: split into simpler fractions.
Long Division
Degree of numerator \( \geq \) degree of denominator: divide first, then integrate.

Core Rule: \( u \)-Substitution

When to use it: The integrand looks like a composite function multiplied by the derivative of the inner function (possibly off by a constant).

Spot inner function Set
\( u = g(x) \)
Integrate
Check: is \(g'(x)\) present (up to a constant)?

Core Rule: Integration by Parts

When to use it: The integrand is a product of two different-type functions (e.g., polynomial times exponential, polynomial times trig, polynomial times logarithm).

Substitution vs. By Parts

\( u \)-Substitution By Parts
Pattern Composition \( f(g(x))\cdot g'(x) \) Product of different types
Key signal Derivative of inner function is present Two "unrelated" factors
Example \( \int 2x\cos(x^2)\,dx \) \( \int x\cos x\,dx \)
Result \( \sin(x^2) + C \) \( x\sin x + \cos x + C \)

Partial Fractions & Long Division

Partial Fractions
Use when the integrand is a proper rational function and the denominator can be factored into linear or irreducible quadratic factors. Decompose, then integrate each term.
Long Division
Use when the degree of the numerator \( \geq \) degree of the denominator. Divide first to obtain a polynomial plus a proper fraction, then integrate each piece.

The Decision Flowchart

Given: \( \int f(x)\,dx \)
Is it a rational function \( \frac{P(x)}{Q(x)} \)?
Yes
\( \deg(P) \geq \deg(Q) \)?
Yes Long Div No Partial Frac No
Composition with \( g'(x) \) present?
Yes u-Sub No By Parts Always try the simplest technique first. Some integrals require combining methods!

How to Select a Technique

1
Simplify first. Expand, factor, or use trig identities to reveal a simpler form.
2
Check for a direct antiderivative. Is the integrand a basic form you know (power rule, \( e^x \), \( \sin x \), etc.)?
3
Look for a chain-rule structure. If you see \( f(g(x))\cdot g'(x) \), use \( u \)-substitution.
4
Product of different types? Use integration by parts (LIATE for choosing \( u \)).
5
Rational function? Use long division (if improper) then partial fractions.

Example 1: \( u \)-Substitution

1
Identify the inner function: \( g(x) = x^3 \). Notice \( g'(x) = 3x^2 \) appears as a factor.
2
Set \( u = x^3 \), so \( du = 3x^2\,dx \). The integral becomes \( \displaystyle\int \cos u\,du \).
3
Integrate: \( \sin u + C = \sin(x^3) + C \).

Example 2: Integration by Parts

1
No chain-rule pattern: \( x \) and \( e^x \) are different types. Use by parts.
2
LIATE: Algebraic before Exponential. Set \( u = x \) (so \( du = dx \)) and \( dv = e^x\,dx \) (so \( v = e^x \)).
3
Apply formula: \( uv - \int v\,du = x\,e^x - \int e^x\,dx = x\,e^x - e^x + C \).

Example 3: Partial Fractions

1
Factor denominator: \( x^2 - 1 = (x-1)(x+1) \). Proper rational function: use partial fractions.
2
Decompose: \( \frac{5x+1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1} \). Solving gives \( A = 3 \), \( B = 2 \).
3
Integrate each term: \( 3\ln|x-1| + 2\ln|x+1| + C \).

Example 4: Combining Techniques

1
Improper fraction: \( \deg(x^3) = 3 \geq \deg(x^2+1) = 2 \). Perform long division first.
2
Divide: \( \frac{x^3}{x^2+1} = x - \frac{x}{x^2+1} \).
3
Integrate each piece: \( \int x\,dx = \frac{x^2}{2} \). For \( \int \frac{x}{x^2+1}\,dx \), use \( u \)-sub with \( u = x^2+1 \): gives \( \frac{1}{2}\ln(x^2+1) \).

Common Mistakes

✕ Forgetting the constant factor
When \( g'(x) \) differs from the integrand by a constant, you must multiply and divide by that constant. E.g., \( \int x\,e^{x^2}dx \) needs a factor of \( \frac{1}{2} \).
✕ Wrong LIATE choice
Choosing \( u = e^x \) instead of \( u = x \) in \( \int x\,e^x\,dx \) makes the integral harder, not simpler. LIATE keeps you on track.
✕ Skipping long division
Applying partial fractions to an improper fraction without dividing first gives wrong decomposition. Always check degrees first.

Technique Summary

Technique Signal Key Step
\( u \)-Sub \( f(g(x))\cdot g'(x) \) Let \( u = g(x) \)
By Parts Product of different types LIATE to pick \( u \)
Partial Fractions Proper rational, factorable denom Decompose, then \( \int \frac{A}{x-a}\,dx \)
Long Division \( \deg(P) \geq \deg(Q) \) Divide first, then integrate

Question 1 of 5

Which technique is most appropriate for evaluating \( \displaystyle\int x^2\,\ln x\,dx \)?

Question 2 of 5

Evaluate \( \displaystyle\int \frac{2x}{x^2 + 9}\,dx \).

Question 3 of 5

What is the first step to evaluate \( \displaystyle\int \frac{x^3 + 2}{x + 1}\,dx \)?

Question 4 of 5

Using integration by parts on \( \displaystyle\int e^x\sin x\,dx \), how many applications are needed before the original integral reappears?

Question 5 of 5

Which integral is best suited for partial fraction decomposition?