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Master U-Substitution: Simplify Complex Integrals

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

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Area Under Curves

Areas between curves

Average Value

Average value of a function

Basic Integration Techniques

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Concept of Limits

Finding limits from graphs

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits

Continuity

Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

Discontinuities of Rational Functions (denominator=0)

Critical Points and Extrema

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Optimization

Curve Analysis

Curve sketching

Derivative Concept

Estimating Derivatives from a table

Slope and equation of tangent line

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Derivative Rules

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivatives of Special Functions

Derivative of inverse trigonometric functions

Derivative of trigonometric functions

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

Fundamental Theorem of Calculus

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Implicit Differentiation

Implicit differentiation

Limits at Infinity

Infinite limits - vertical asymptotes

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Motion and Rates

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Riemann Sums

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

Tangent Lines

Linear approximation

Use of Linear Approximations in Estimating Square Roots

Linearization of Radical Functions

Linearization of Polynomial Functions

Linearization of Rational Functions

Linearization of Exponential Functions

Linearization of Logarithmic Functions

Linear Approximation of Trigonometric Functions

Parent Assignments

Challenge Zone

Foundation Review

math

Intro

Intro to U-Substitution with a trigonometric integral example

Integrating a polynomial raised to a power using u-substitution

Integrating e to the power of 2x using u-substitution

Definite integral of sine over cosine squared using u-substitution

Why u-substitution fails when the integrand has a discontinuity

U-substitution with a cube root and polynomial expression

Integrating a square root function using u-substitution

Integrating ln(x) cubed over x using u-substitution

Integrating 1/(x ln x) using u-substitution with natural log

Integrating sin³(x)cos(x) using u-substitution

Integrating (tan x - 1)^100 · sec²x using u-substitution

U-substitution with x^5 times sqrt(x^3 - a) by splitting exponents

Integrating x^5 times cube root of (1+x^2) using u-substitution

Integrating (1+x)/(1+x²) by splitting the fraction and u-substitution

Integrating cotangent x using trig identity and u-substitution

Comparing indefinite vs definite integrals and their geometric meaning

Changing bounds of integration when using u-substitution

Comparing the two approaches for evaluating definite integrals with u-sub

Evaluating a definite integral using u-substitution (Approach 1)

You may start to notice that some integrals cannot be integrated by normal means. Therefore, we introduce a method called <a href="[[glossary.html|{"keyword":"u-Substitution"}]]">U-Substitution</a>. This method involves substituting ugly functions as the letter "u", and therefore making our integrands easier to integrate. We will use this technique to integrate many different functions such as polynomial functions, irrational functions, trigonometric functions, exponential functions and logarithmic functions. We will also integrate functions with a combination of different types of functions.

How to use u-substitution to find integrals

U-Substitution: The Key to Solving Complex Integrals

What You'll Learn

Identify when normal integration fails and u-substitution is required

Select the appropriate function to substitute as u using pattern recognition

Transform integrals by replacing complex expressions with simpler variable u

Apply u-substitution to polynomial, trigonometric, exponential, and logarithmic functions

Convert definite integral bounds from x-values to corresponding u-values

Verify continuity requirements before evaluating definite integrals

What You'll Practice

Substituting composite functions and matching derivatives in integrands

Integrating radical, rational, and trigonometric expressions using u-substitution

Evaluating definite integrals by converting bounds to u-values

Identifying and avoiding discontinuities in definite integrals

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

19 Video lessons

Practice exercises

Learning resources

Skills

U-Substitution

Integration Techniques

Definite Integrals

Composite Functions

Chain Rule

Continuity

Antiderivatives

Variable Transformation