u-Substitution

u-Substitution

Part 1 of 7 — Basic u-Substitution

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

How to Choose $u$

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

Worked Examples

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.

Basic u-Substitution 🎯

The Linear Substitution Shortcut

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

$\boxed{\int f(ax+b)\,dx = \frac{1}{a}F(ax+b) + C}$

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

Choose the right $u$ for each integral. 🔍

Compute the integral. ✍️

Key Takeaways — Part 1

Up Next: Part 2 — Adjusting for Missing 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.

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

Worked Example

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

$u = x^3$, $du = 3x^2\,dx$. 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

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

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

Up Next: 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:

$\boxed{\int_a^b f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du}$

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:

Common Mistakes When Changing Limits

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 $[0,\pi]$.

Side-by-Side Comparison

Even/Odd Shortcuts with u-Sub

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

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

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

Up Next: 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.

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

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

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

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

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

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

Logarithmic Substitutions

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

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

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

Up Next: Part 5 — Long Division & 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:

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

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:

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

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

Up Next: Part 6 — Problem-Solving Workshop.

u-Substitution

Part 6 of 7 — Problem-Solving Workshop

Integration Strategy Flowchart

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

Classify each integral. 🔍

Mixed practice. ✍️

Key Takeaways — Part 6

Up Next: Part 7 — Comprehensive Assessment.

u-Substitution

Part 7 of 7 — Comprehensive Assessment

Complete Formula Reference

Top AP Mistakes to Avoid

AP-Style Questions — Set 1 🎯

AP-Style Questions — Set 2 🎯

Final classification challenge. 🔍

Final challenge. ✍️

u-Substitution — Complete Summary

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