🎯⭐ INTERACTIVE LESSON

u-Substitution

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u-Substitution - Complete Interactive Lesson

Part 1: Basic u-Substitution

u-Substitution

Part 1 of 7 — Basic u-Substitution

Part	Topic
1	Basic u-Substitution
2	Adjusting for Missing Constants
3	Definite Integrals with u-Sub
4	Trickier Substitutions
5	Long Division & Completing the Square
6	Problem-Solving Workshop
7	Comprehensive Review

The Big Idea

$\boxed{\text{u-Substitution is the REVERSE of the Chain Rule}}$

The Chain Rule says: $\frac{d}{dx}[F(g(x))] = F'(g(x)) \cdot g'(x) = f(g(x)) \cdot g'(x)$ .

Working backwards: $\int f(g(x)) \cdot g'(x)\,dx = F(g(x)) + C$ .

The 5-Step Method

Step	Action	Example: $\int 2x\cos(x^2)\,dx$
1	Choose $u$ = inner function	$u = x^2$

How to Choose $u$

$\boxed{u = \text{the inner function of a composition}}$

Look For	Choose $u$	Why

Basic u-Substitution 🎯

The Linear Substitution Shortcut

For $\int f(ax + b)\,dx$ where $f$ has a known antiderivative $F$ :

Choose the right $u$ for each integral. 🔍

Compute the integral. ✍️

Key Takeaways — Part 1

Concept	Key Rule
u-Sub = reverse Chain Rule	$\int f(g(x))g'(x)\,dx = F(g(x)) + C$

Part 2: Adjusting for Constants

u-Substitution

Part 2 of 7 — Adjusting for Missing Constants

When $du$ Doesn't Match Exactly

Often the constant coefficient doesn't match. You can multiply and divide by constants to fix this.

$\boxed{\int f(g(x)) \cdot k \cdot g'(x)\,dx = k \int f(u)\,du}$

Part 3: Definite Integrals with u-Sub

u-Substitution

Part 3 of 7 — Definite Integrals with u-Substitution

Two Approaches

When evaluating a definite integral with u-substitution, you have two choices:

Method	Steps	When to Use
Change the limits	Convert everything to $u$ , including bounds	Cleaner — preferred on AP Exam
Back-substitute	Find antiderivative in terms of $x$ , then evaluate	When limits are easy to convert back

$\int_{a}^{b} f ($

Part 4: Trickier Substitutions

u-Substitution

Part 4 of 7 — Trickier Substitutions

Beyond Basic Patterns

Some integrals require creative choices of $u$ or algebraic manipulation before substitution.

Category	Example	Strategy
Exponential inside	$\int \frac{e^x}{1+e^x}\,dx$

Part 5: Long Division and Completing the Square

u-Substitution

Part 5 of 7 — Long Division & Completing the Square

Algebraic Manipulation Before Integrating

Some rational functions or quadratics need algebraic prep work BEFORE substitution. Two key techniques:

Technique	When to Use	Goal
Long division	Degree of numerator $\geq$ degree of denominator	Reduce to polynomial $+$ proper fraction
Completing the square	Irreducible quadratic in denominator	Create $\arctan$ or form

Part 6: Problem-Solving Workshop

u-Substitution

Part 6 of 7 — Problem-Solving Workshop

Integration Strategy Flowchart

Step	Question	Action
1	Is it a known basic form?	$\int x^n, e^x, \sin x$ , etc. — integrate directly

Part 7: Comprehensive Assessment

u-Substitution

Part 7 of 7 — Comprehensive Assessment

Complete Formula Reference

Integral	Formula
$\int [f(x)]^n f'(x)\,dx$

Example 1: $\int 3x^2 e^{x^3}\,dx$

$u = x^3$ , $du = 3x^2\,dx$ → $\int e^u\,du = e^{x^3} + C$

Example 2: $\int \frac{5}{(5x+1)^3}\,dx$

$u = 5x+1$ , $du = 5\,dx$ → $\int u^{-3}\,du = -\frac{1}{2}u^{-2} + C = -\frac{1}{2(5x+1)^2} + C$

Example 3: $\int x\sqrt{x^2+1}\,dx$

$u = x^2+1$ , $du = 2x\,dx$ , so $x\,dx = \frac{du}{2}$

$\frac{1}{2}\int u^{1/2}\,du = \frac{1}{2} \cdot \frac{2}{3}u^{3/2} = \frac{1}{3}(x^2+1)^{3/2} + C$

AP Tip: Always check your answer by differentiating! If $\frac{d}{dx}[\text{answer}] = \text{integrand}$ , you're correct.

\int f (a x + b) d x = \frac{1}{a} F (a x + b) + C

Integral	Result	Using $a =$
$\int \sin(5x)\,dx$	$-\frac{1}{5}\cos(5x) + C$	$a = 5$
$\int e^{-2x}\,dx$	$-\frac{1}{2}e^{-2x} + C$
$\int (3x+7)^4\,dx$	$\frac{(3x+7)^5}{15} + C$
$\int \sec^2(4x)\,dx$	$\frac{1}{4}\tan(4x) + C$

Key Fact: This shortcut ONLY works for LINEAR inner functions ( $ax + b$ ). For nonlinear inner functions, you must do the full u-sub process.

Up Next: Part 2 — Adjusting for Missing Constants.

$\int x^2 e^{x^3}\,dx$

$u = x^3$ , $du = 3x^2\,dx$ . We have $x^2\,dx$ but need $3x^2\,dx$ :

$\frac{1}{3}\int 3x^2 e^{x^3}\,dx = \frac{1}{3}\int e^u\,du = \frac{e^{x^3}}{3} + C$

Key Fact: You can ONLY move constants outside the integral. You can NEVER move a variable ( $x$ , $t$ , etc.) outside!

Essential Patterns to Recognize

Pattern	Choose $u$	Result
$\int f(ax+b)\,dx$	$u = ax+b$	$\frac{1}{a}F(ax+b) + C$
$\int x^{n-1} f(x^n)\,dx$	$u = x^n$
$\int f(\sin x)\cos x\,dx$	$u = \sin x$	$F(\sin x) + C$
$\int f(\cos x)\sin x\,dx$	$u = \cos x$	$-F(\cos x) + C$
$\int f(\tan x)\sec^2 x\,dx$	$u = \tan x$
$\int f(e^x)e^x\,dx$	$u = e^x$
$\int f(\ln x)\frac{1}{x}\,dx$	$u = \ln x$
$\int \frac{f'(x)}{f(x)}\,dx$

The $\frac{f'}{f} \to \ln$ Pattern

This is one of the most important patterns:

$\boxed{\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C}$

Example: $\int \tan x\,dx = \int \frac{\sin x}{\cos x}\,dx$

$u = \cos x$ , $du = -\sin x\,dx$ : $-\int \frac{du}{u} = -\ln|\cos x| + C = \ln|\sec x| + C$

Pattern Recognition 🎯

What u-Sub CANNOT Do

Invalid Operation	Why It Fails
Move $x$ outside: $x\int \ldots$	Variables aren't constant!
$\int e^{x^2}\,dx$	No elementary antiderivative exists
$\int \sin(x^2)\,dx$	No closed form (Fresnel integral)
Product of unrelated functions	u-sub needs $f(g(x)) \cdot g'(x)$ pattern

AP Tip: If you can't find a $u$ that works, the problem might require a different technique (rewriting, inverse trig, etc.) or the integrand might already be in a recognized form.

Identify the pattern. 🔍

Apply the pattern. ✍️

Key Takeaways — Part 2

Concept	Key Rule
Constant adjustment	Multiply/divide by constants to match $du$
$f'/f$ pattern	$\int \frac{f'}{f} = \ln
CANNOT move variables outside	Only constants can come out of $\int$
No elementary form	Some integrals (like $e^{x^2}$ ) have no closed form

Up Next: Part 3 — Definite Integrals with u-Sub.

\int_{a}^{b} f (g (x)) g^{'} (x) d x = \int_{g (a)}^{g (b)} f (u) d u

Key Fact: When you change limits, you NEVER need to convert back to $x$ . Evaluate directly in $u$ .

Method 1: Change the Limits (Recommended)

Step-by-step process:

Step	Action	Example: $\int_0^2 x(x^2+1)^3\,dx$
1	Choose $u$	$u = x^2+1$
2	Find $du$

Common Mistakes When Changing Limits

Mistake	Problem	Fix
Keeping old limits	$\int_0^2 u^3\,du$ is WRONG	Convert: $\int_{1}^{5}^{}$

Practice with changing limits. 🎯

Method 2: Back-Substitute

With this method, you integrate in $u$ , convert back to $x$ , then evaluate with original limits.

Example: $\int_0^{\pi} \sin^2(x)\cos(x)\,dx$

$u = \sin x$ , $du = \cos x\,dx$ :

$\int u^2\,du = \frac{u^3}{3} + C$

Back-substitute: $\frac{\sin^3(x)}{3}\Big|_0^{\pi} = \frac{\sin^3(\pi)}{3} - \frac{\sin^3(0)}{3} = 0 - 0 = 0$

AP Tip: The answer being $0$ makes sense! $\sin^2(x)\cos(x)$ is an odd function about $x = \pi/2$ on .

Side-by-Side Comparison

Feature	Change Limits	Back-Substitute
Convert bounds?	Yes, to $u$	No
Convert back to $x$ ?	No	Yes
Risk of errors	Slightly lower	Slightly higher
Speed	Usually faster	May be slower

Even/Odd Shortcuts with u-Sub

$\boxed{\text{If } f \text{ is even: } \int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx}$

$\boxed{\text{If } f \text{ is odd: } \int_{-a}^a f(x)\,dx = 0}$

Integrand	Even/Odd	$\int_{-a}^a$
$x^2 e^{-x^2}$

Key Fact: Spotting symmetry on the AP Exam can save significant time and avoid messy computations.

Choose the best approach. 🔍

Evaluate a definite integral. ✍️

Key Takeaways — Part 3

Concept	Key Rule
Change limits (preferred)	$\int_a^b \to \int_{g(a)}^{g(b)}$ , evaluate in $u$
Back-substitute	Integrate in $u$ , convert to $x$ , use original limits
Even function shortcut	$\int_{-a}^a f = 2\int_0^a f$
Odd function shortcut	$\int_{-a}^a f = 0$

Up Next: Part 4 — Trickier Substitutions.

Exponential Substitutions

$\boxed{\int \frac{e^x}{f(e^x)}\,dx: \text{ let } u = e^x \text{ or } u = f(e^x)}$

Example 1: $\int \frac{e^{2x}}{e^x + 3}\,dx$

Let $u = e^x + 3$ , $du = e^x\,dx$ . Rewrite :

$\int \frac{e^x \cdot e^x}{e^x + 3}\,dx = \int \frac{u - 3}{u}\,du = \int \left(1 - \frac{3}{u}\right)du = u - 3\ln|u| + C = e^x + 3 - 3\ln(e^x+3) + C$

Example 2: $\int e^x \sqrt{e^x + 1}\,dx$

$u = e^x + 1$ , $du = e^x\,dx$ :

$\int \sqrt{u}\,du = \frac{2}{3}u^{3/2} + C = \frac{2}{3}(e^x+1)^{3/2} + C$

Key Fact: When $e^x$ appears both in the integrand and provides $du$ , try $u = \text{(the other piece)}$ .

Trickier integrals. 🎯

Square Root Substitutions — Solving for $x$

When $x$ appears outside the composition, express $x$ in terms of $u$ .

Example: $\int x^2\sqrt{x+2}\,dx$

$u = x+2$ , so $x = u-2$ , $x^2 = (u-2)^2 = u^2 - 4u + 4$ , :

$\int (u^2-4u+4)\sqrt{u}\,du = \int (u^{5/2} - 4u^{3/2} + 4u^{1/2})\,du$

$= \frac{2u^{7/2}}{7} - \frac{8u^{5/2}}{5} + \frac{8u^{3/2}}{3} + C$

Substitute back: replace $u$ with $x+2$ .

Step	Purpose
Choose $u$ = expression under radical	Simplifies the root
Express all other $x$ 's as $u \pm c$	Everything becomes a $u$ -integral
Expand and integrate term by term	Standard power rule
Back-substitute

Logarithmic Substitutions

When $\ln x$ appears as an argument and $1/x$ is present (or producible):

$\boxed{u = \ln x, \quad du = \frac{1}{x}\,dx}$

Integral	Substitution	Result
$\int \frac{(\ln x)^n}{x}\,dx$

AP Tip: The combination " $\frac{\text{stuff with } \ln x}{x}$ " almost always signals $u = \ln x$ .

Choose the correct substitution. 🔍

Try a tricky one. ✍️

Key Takeaways — Part 4

Technique	When to Use
$u = e^x + c$	Exponential in denominator or under root
$u = \ln x$	$\ln x$ in numerator with $1/x$ present
$u =$ radical expression	Solve for $x = f(u)$ and substitute
Nested: $u =$ inner function	$1/x$ or $e^x$ provides $du$

Up Next: Part 5 — Long Division & Completing the Square.

Long Division for Improper Rational Functions

$\boxed{\text{If } \deg(\text{num}) \geq \deg(\text{denom}), \text{ divide first!}}$

Example: $\int \frac{x^2+1}{x+1}\,dx$

Perform long division: $\frac{x^2+1}{x+1} = x - 1 + \frac{2}{x+1}$

$\int \left(x - 1 + \frac{2}{x+1}\right)dx = \frac{x^2}{2} - x + 2\ln|x+1| + C$

Quick Division Patterns

Fraction	After Division	Integral
$\frac{x^2}{x+1}$	$x - 1 + \frac{1}{x+1}$

AP Tip: If you see $\frac{\text{bigger polynomial}}{\text{smaller polynomial}}$ , long division is almost certainly the first step.

Long division practice. 🎯

Completing the Square

When the denominator is a quadratic that doesn't factor nicely:

$\boxed{ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)}$

Why? This creates forms you know:

Resulting Form	Integral
$\frac{1}{u^2 + k^2}$

Worked Example

$\int \frac{1}{x^2 + 6x + 13}\,dx$

Complete the square: $x^2 + 6x + 13 = (x+3)^2 + 4$

$\int \frac{1}{(x+3)^2 + 4}\,dx$

$u = x+3$ , $du = dx$ :

$\int \frac{du}{u^2 + 2^2} = \frac{1}{2}\arctan\frac{u}{2} + C = \frac{1}{2}\arctan\frac{x+3}{2} + C$

More Completing the Square Examples

Example 2: $\int \frac{1}{\sqrt{3-2x-x^2}}\,dx$

Rewrite: $3-2x-x^2 = -(x^2+2x-3) = -(x^2+2x+1-4) = 4-(x+1)^2$

$\int \frac{dx}{\sqrt{4-(x+1)^2}} = \arcsin\frac{x+1}{2} + C$

Decision Guide

Denominator	Complete the Square?	Result Type
$x^2 + bx + c$ (positive leading coeff, no real roots)	Yes	$\arctan$ form
$c - b$ (under square root, positive)

Key Fact: If the quadratic has real roots, it factors — use partial fractions (BC topic). If it has no real roots, complete the square for $\arctan$ .

Identify the technique. 🔍

Combine techniques. ✍️

Key Takeaways — Part 5

Technique	When	Result
Long division	$\deg(\text{num}) \geq \deg(\text{denom})$	Polynomial $+$ simple fraction
Complete the square	Irreducible quadratic	$\arctan$ or $\arcsin$ form
Factor & partial fractions	Reducible quadratic (BC only)	Sum of $\ln$ terms

Up Next: Part 6 — Problem-Solving Workshop.

Worked Examples — Full Solutions

Example 1: $\int \frac{x^3}{x^2+1}\,dx$

Long division: $\frac{x^3}{x^2+1} = x - \frac{x}{x^2+1}$

$\int x\,dx - \int \frac{x}{x^2+1}\,dx = \frac{x^2}{2} - \frac{1}{2}\ln(x^2+1) + C$

For the second integral: $u = x^2+1$ , $du = 2x\,dx$ .

Example 2: $\int \frac{\sin x}{1 + \cos^2 x}\,dx$

$u = \cos x$ , $du = -\sin x\,dx$ :

$-\int \frac{du}{1+u^2} = -\arctan(\cos x) + C$

Example 3: $\int x e^{x^2}\,dx$

$u = x^2$ , $du = 2x\,dx$ :

$\frac{1}{2}\int e^u\,du = \frac{e^{x^2}}{2} + C$

Example 4: $\int \frac{e^x - e^{-x}}{e^x + e^{-x}}\,dx$

Recognize $f'/f$ pattern: the numerator IS the derivative of the denominator!

$= \ln|e^x + e^{-x}| + C = \ln(e^x + e^{-x}) + C$

AP Tip: This is $\int \tanh x\,dx$ (hyperbolic tangent). The $f'/f$ pattern saves enormous work.

Classify and solve. 🎯

Common AP Exam Mistakes

Mistake	Example	Correct Approach
Moving $x$ outside integral	" $x \int \cos(x^2)\,dx$ "	Use u-sub: $u=x^2$
Forgetting to change limits	$\int_0^1 \to \int_0^1$ in $u$
Wrong constant factor	$du = 2x\,dx$ , forgetting $1/2$	Track adjustment carefully
Not simplifying first	Integrating $\frac{x^3+x}{x}$ as-is	Simplify to $x^2+1$ first
Mixing up $+C$ on definite	Adding $+C$ to $\int_a^b$	No $+C$ for definite integrals
Wrong sign on $\cos x$ sub	$u = \cos x$ , $du = \sin x\,dx$	$d u$ (negative!)

Classify each integral. 🔍

Mixed practice. ✍️

Key Takeaways — Part 6

Problem Type	First Step
Improper fraction	Long division
Factorable numerator/denominator	Cancel common factors
Composite function with derivative present	u-substitution
Irreducible quadratic	Complete the square
$f'/f$ pattern	$\ln
Linear argument	Shortcut: $\frac{1}{a}F(ax+b)$

Up Next: Part 7 — Comprehensive Assessment.

Top AP Mistakes to Avoid

Mistake	Correction
Moving variables outside $\int$	Only CONSTANTS can exit
Forgetting $\frac{1}{a}$ in linear shortcut	Always divide by inner coefficient
Wrong sign: $d(\cos x) = -\sin x\,dx$	Track negative signs carefully
Not changing limits with $u$ -sub	Convert: $x \to u(x)$
Adding $+C$ to definite integrals	No $+C$ when bounds are present
Forgetting to back-substitute	If using original limits, convert $u \to x$

AP-Style Questions — Set 1 🎯

AP-Style Questions — Set 2 🎯

Final classification challenge. 🔍

Final challenge. ✍️

u-Substitution — Complete Summary

Concept	Key Idea
Basic u-sub	Reverse chain rule: $\int f(g(x))g'(x)\,dx$
Constant adjustment	Multiply/divide by constants (NEVER variables)
Definite integrals	Change limits to $u$ -values, or back-substitute
Linear shortcut	$\int f(ax+b)\,dx = \frac{1}{a}F(ax+b)+C$
$f'/f$ pattern	$\int f'/f = \ln
Trickier subs	Express extra $x$ 's in terms of $u$
Long division	When $\deg(\text{num}) \geq \deg(\text{denom})$
Complete the square	Irreducible quadratic $\to \arctan/\arcsin$
Symmetry	Odd integrand on $[-a,a] \to 0$

Congratulations! You've mastered u-substitution — the most important integration technique on the AP Calculus AB exam.

- \int \frac{d u}{u} = - ln ∣ cos x ∣ + C = ln ∣ sec x ∣ + C

d u = 2 x d x \Rightarrow x d x = \frac{d u}{2}

\int_{1}^{5} u^{3} d u

e^{2x} = e^x \cdot e^x

e^{2 x} = e^{x} \cdot e^{x}

x - 1 + \frac{1}{x + 1}

\frac{1}{k}\arctan\frac{u}{k} + C

\frac{1}{k} arctan \frac{u}{k} + C

$(\text{stuff})^n$ with its derivative nearby	$u = \text{stuff}$	Power of a function
$e^{\text{stuff}}$ with derivative of stuff	$u = \text{stuff}$	Exponential compound
$\sin(\text{stuff})$ , $\cos(\text{stuff})$	$u = \text{stuff}$	Trig compound
$\ln(\text{stuff})$ in denominator	$u = \text{stuff}$	Log pattern

u-Substitution

u-Substitution - Complete Interactive Lesson

Part 1: Basic u-Substitution

u-Substitution

The Big Idea

The 5-Step Method

How to Choose uuu

The Linear Substitution Shortcut

Key Takeaways — Part 1

Part 2: Adjusting for Constants

u-Substitution

When dududu Doesn't Match Exactly

Part 3: Definite Integrals with u-Sub

u-Substitution

Two Approaches

Part 4: Trickier Substitutions

u-Substitution

Beyond Basic Patterns

Part 5: Long Division and Completing the Square

u-Substitution

Algebraic Manipulation Before Integrating

Part 6: Problem-Solving Workshop

u-Substitution

Integration Strategy Flowchart

Part 7: Comprehensive Assessment

u-Substitution

Complete Formula Reference

Worked Examples

Worked Example

Essential Patterns to Recognize

The f′f→ln⁡\frac{f'}{f} \to \lnff′​→ln Pattern

What u-Sub CANNOT Do

Key Takeaways — Part 2

Method 1: Change the Limits (Recommended)

Common Mistakes When Changing Limits

Method 2: Back-Substitute

Side-by-Side Comparison

Even/Odd Shortcuts with u-Sub

Key Takeaways — Part 3

Exponential Substitutions

Square Root Substitutions — Solving for xxx

Logarithmic Substitutions

Key Takeaways — Part 4

Long Division for Improper Rational Functions

Quick Division Patterns

Completing the Square

Worked Example

More Completing the Square Examples

Decision Guide

Key Takeaways — Part 5

Worked Examples — Full Solutions

Common AP Exam Mistakes

Key Takeaways — Part 6

Top AP Mistakes to Avoid

u-Substitution — Complete Summary

How to Choose $u$

When $du$ Doesn't Match Exactly

The $\frac{f'}{f} \to \ln$ Pattern

Square Root Substitutions — Solving for $x$