Learn advanced calculus techniques through comprehensive video lessons covering integration by substitution, inverse trigonometric functions, and their applications. Master calculating both definite and indefinite integrals using substitution methods, including complex examples involving trigonometric identities and exponential functions. Explore the differentiation and integration of inverse trigonometric functions, including detailed proofs for derivatives of inverse sine, cosine, and tangent functions. Practice applying substitution techniques to challenging problems, working with hidden conditions, domain restrictions, and optimization scenarios. Develop skills in evaluating definite integrals through substitution, interpreting inverse functions, and using trigonometric identities to simplify complex integration problems. Build proficiency in recognizing when and how to apply various substitution methods, from simple u-substitutions to more sophisticated trigonometric substitutions for functions involving square roots and inverse trigonometric expressions.

Syllabus

Calculating Definite Integrals by Substitution
Calculating Indefinite Integrals by Substitution (1 of 2: Integrating)
Calculating Indefinite Integrals by Substitution (2 of 2: Resolving θ)
Definite Integrals by Substitution
Integration by Substitution (1 of 2)
Integration by Substitution (2 of 2)
Integrating Trigonometric Function w/ Substitution & Identity
Determining & Interpreting the Inverse of a Special Exponential Function
Evaluating Derivative of an Inverse Function
Differentiation of ITFs (1 of 3: Predicting the properties of d/dx(tan¯¹(x)))
Differentiation of Inverse Trig Functions (1 of 2: Dealing with a Hidden Condition in the Question)
Differentiation of ITFs (2 of 3: Using Trig Identities to find the derivative of tan¯¹(x))
Differentiation of ITFs (3 of 3: Finding the Condition that make d/dx(sin¯¹(x)) what it actually is)
Differentiation of Inverse Trig Functions (2 of 2: Maximum Optimisation for Inverse Trig Function)
Calculus with ITFs (1 of 3: Generalising the derivative of the Basic Inverse sine function)
Calculus with ITFs (2 of 3: Generalising the derivative of tan¯¹(x) & the derivative of cos¯¹(x))
Calculus with ITFs (3 of 3: Integrating cos¯¹(x) to find the area under a set domain)
Integration by Substitution (1 of 4: Integrating (1-4x^2)^1/2 without Substitution)
Integration by Substitution (2 of 4: Doing the same Question with a simple substitution of 'u')
Integration by Substitution (3 of 4: Applying the substitution on a harder example)
Integration by Substitution (4 of 4: Applying the substitution to a definite integral)
Harder Integration by Substitution (1 of 3: Substituting a tan function to simplify the Integration)
Harder Integration by Substitution (2 of 3: Dealing with implied restrictions within the Question)
Harder Integration by Substitution (3 of 3: Using another substitution to find the solution)
Properties of Definite Integrals (6 of 6: Using Integration by substitution to evaluate integral)
Differentiating Inverse Trigonometric Functions (3 of 6: Examples + graphing the derivative)
Differentiating Inverse Trigonometric Functions (4 of 6: Proof for inverse sine)
Differentiating Inverse Trigonometric Functions (5 of 6: Graphing inverse sine derivative)
Differentiating Inverse Trigonometric Functions (2 of 6: Proof for inverse tan)
Differentiating Inverse Trigonometric Functions (6 of 6: Cosine)
Differentiating Inverse Trigonometric Functions (1 of 6: Laying foundations)
Integrating into Inverse Trigonometric Functions
Integration by Substitution (1 of 2: Indefinite integrals)
Integration by Substitution (2 of 2: Definite integrals)