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Integrating Rational Polynomials (Example)

Integrating Rational Polynomials (Example)

Partial Fractions (example with cubic denominator)

Partial Fractions (example with cubic denominator)

Partial Fractions for Integrating Rational Polynomials

Partial Fractions for Integrating Rational Polynomials

Partial Fractions w/ Perfect Square Denominators

Partial Fractions w/ Perfect Square Denominators

Rearranging Rational Polynomials w/ Division Transformation

Rearranging Rational Polynomials w/ Division Transformation

Deriving Integration by Parts

Deriving Integration by Parts

Integration by Parts - Example 1: x cos x

Integration by Parts - Example 1: x cos x

Integration by Parts - Example 2: x*e^x & x²*e^x

Integration by Parts - Example 2: x*e^x & x²*e^x

Integration by Parts - Example 3: ln x

Integration by Parts - Example 3: ln x

Integration by Parts - Example 4: e^x * sin x

Integration by Parts - Example 4: e^x * sin x

An unusual integral with two very different solutions?

An unusual integral with two very different solutions?

Integration by Parts - Example 5: sin¯¹(x)

Integration by Parts - Example 5: sin¯¹(x)

Integrals that lead to Logarithmic Functions

Integrals that lead to Logarithmic Functions

Introduction to Recurrence Relations

Introduction to Recurrence Relations

Recurrence Relation Example: Integrating (sin x)^n

Recurrence Relation Example: Integrating (sin x)^n

Recurrence Relations without IBP (1 of 2): (tan x)^n

Recurrence Relations without IBP (1 of 2): (tan x)^n

Recurrence Relations without IBP (2 of 2): (x^n)/(x+1)

Recurrence Relations without IBP (2 of 2): (x^n)/(x+1)

Sometimes integration by parts is a bad idea...

Sometimes integration by parts is a bad idea...

Integrating with t-results (1 of 3): Example integral through identities

Integrating with t-results (1 of 3): Example integral through identities

Integrating with t-results (2 of 3): Changing the variable of integration

Integrating with t-results (2 of 3): Changing the variable of integration

Integrating with t-results (3 of 3): Further example

Integrating with t-results (3 of 3): Further example

Why do logarithmic functions have absolute value signs after integration?

Why do logarithmic functions have absolute value signs after integration?

Integrating √[x/(1-x)] by Trigonometric Substitution

Integrating √[x/(1-x)] by Trigonometric Substitution

Correction of Wrong Solution from Fitzpatrick (integration question)

Correction of Wrong Solution from Fitzpatrick (integration question)

Tricky Recurrence Relation with Unexpected Algebraic Manipulation

Tricky Recurrence Relation with Unexpected Algebraic Manipulation

Evaluating Recurrence Relations

Evaluating Recurrence Relations

Proving Properties of Definite Integrals (1 of 2)

Proving Properties of Definite Integrals (1 of 2)

Proving Properties of Definite Integrals (2 of 2)

Proving Properties of Definite Integrals (2 of 2)

3 Definite Integrals Evaluated by Trigonometric Substitution

3 Definite Integrals Evaluated by Trigonometric Substitution

Integrating √(16-x²)/x² by Trigonometric Substitution

Integrating √(16-x²)/x² by Trigonometric Substitution

Integrating √[(1+x)/(1-x)] by Trigonometric Substitution

Integrating √[(1+x)/(1-x)] by Trigonometric Substitution

Evaluating Definite Integral w/ t-results

Evaluating Definite Integral w/ t-results

Integral of sin(x)tan²(x): Three Approaches

Integral of sin(x)tan²(x): Three Approaches

Proving Inequality from an Increasing Function

Proving Inequality from an Increasing Function

Proving Inequality from Prior Integral

Proving Inequality from Prior Integral

Proving that a function is increasing

Proving that a function is increasing

Using Properties of Definite Integrals: Trigonometric Integrand (1 of 2)

Using Properties of Definite Integrals: Trigonometric Integrand (1 of 2)

Determining Algebraic Recurrence Relation

Determining Algebraic Recurrence Relation

Expressing I(n) without Recursion

Expressing I(n) without Recursion

Using Properties of Definite Integrals: Trigonometric Integrand (2 of 2)

Using Properties of Definite Integrals: Trigonometric Integrand (2 of 2)

Harder (Ext 2) Integral Requiring Substitution

Harder (Ext 2) Integral Requiring Substitution

Harder (Ext 2) Integral Question (1 of 2: Manipulating the Recurrence Relation)

Harder (Ext 2) Integral Question (1 of 2: Manipulating the Recurrence Relation)

Harder (Ext 2) Integral Question (2 of 2: Evaluating Related Limiting Sum)

Harder (Ext 2) Integral Question (2 of 2: Evaluating Related Limiting Sum)

Partial Fractions (1 of 3: Introducing an identity to simplify a quadratic denominator)

Partial Fractions (1 of 3: Introducing an identity to simplify a quadratic denominator)

Partial Fractions (2 of 3: An alternative method to find introduced variables)

Partial Fractions (2 of 3: An alternative method to find introduced variables)

$Partial Fractions (3 of 3: How to use Partial Fractions to split fractions with Quadratic Numerator)$

Partial Fractions (3 of 3: How to use Partial Fractions to split fractions with Quadratic Numerator)

Partial Fractions: Quadratic Factors (1 of 2: Issues with non-linear denominator factors)

Partial Fractions: Quadratic Factors (1 of 2: Issues with non-linear denominator factors)

Partial Fractions: Quadratic Factors (2 of 2: Problems with Quadratic Factors and Solution)

Partial Fractions: Quadratic Factors (2 of 2: Problems with Quadratic Factors and Solution)

$Partial Fraction: Repeated Linear Factors (Technique for Breaking down into partial fractions)$

Partial Fraction: Repeated Linear Factors (Technique for Breaking down into partial fractions)

Integration of Square Root Function (1 of 2: Using a Trig Substitution to help with integration)

Integration of Square Root Function (1 of 2: Using a Trig Substitution to help with integration)

Integration of Square Root Function (2 of 2: What is this kind of integral solving for?)

Integration of Square Root Function (2 of 2: What is this kind of integral solving for?)

Further Integration (1 of 2: Brief Overview of Extension II Integration)

Further Integration (1 of 2: Brief Overview of Extension II Integration)

Further Integration (2 of 2: Integrating Trig Functions without given substitution)

Further Integration (2 of 2: Integrating Trig Functions without given substitution)

Further Integration [Continued] (1 of 3: Using Double Angle & Completing the Square to integrate)

Further Integration [Continued] (1 of 3: Using Double Angle & Completing the Square to integrate)

Further Integration [Continued] (2 of 3: Adding and subtracting a constant to simplify integral)

Further Integration [Continued] (2 of 3: Adding and subtracting a constant to simplify integral)

Further Integration [Continued] (3 of 3: 'Breaking Apart' Integrals to simplify for integration)

Further Integration [Continued] (3 of 3: 'Breaking Apart' Integrals to simplify for integration)

Partial Fractions & Integration (Using Partial Fractions to simplify an integral for evaluation)

Partial Fractions & Integration (Using Partial Fractions to simplify an integral for evaluation)

Harder Reverse Chain Rule Integral (Using a Substitution and Restrictions to evaluate)

Harder Reverse Chain Rule Integral (Using a Substitution and Restrictions to evaluate)

Proving a Result from the Standard Integrals (1 of 3: Differentiate, Hence Integrate Proof)

Proving a Result from the Standard Integrals (1 of 3: Differentiate, Hence Integrate Proof)

Proving a Result from the Standard Integrals (2 of 3: Substituting tan to simplify the integral)

Proving a Result from the Standard Integrals (2 of 3: Substituting tan to simplify the integral)

Proving a Result from the Standard Integrals (3 of 3: Using Partial Fractions to simplify sec)

Proving a Result from the Standard Integrals (3 of 3: Using Partial Fractions to simplify sec)

Integration with t-results (1 of 2: Changing the variable of integration)

Integration with t-results (1 of 2: Changing the variable of integration)

Integration with t-results (2 of 2: Dealing with the integral in t)

Integration with t-results (2 of 2: Dealing with the integral in t)

Integration by Parts (1 of 3: Deriving the Formula)

Integration by Parts (1 of 3: Deriving the Formula)

Integration by Parts (2 of 3: How to choose u & dv)

Integration by Parts (2 of 3: How to choose u & dv)

Integration by Parts (3 of 3: Integral of sin¯¹(x))

Integration by Parts (3 of 3: Integral of sin¯¹(x))

Integration by t results (Explanation of Harder Question)

Integration by t results (Explanation of Harder Question)

Integration by Parts (What to choose for u and dv for integration by parts?)

Integration by Parts (What to choose for u and dv for integration by parts?)

Integration with t Results (1 of 2: Changing the variable to solve with t results)

Integration with t Results (1 of 2: Changing the variable to solve with t results)

Integration with t Results (2 of 2: Simplifying the integral before applying t results)

Integration with t Results (2 of 2: Simplifying the integral before applying t results)

Recurrence Relations (1 of 4: Introduction to Recurrence relations with introductory examples)

Recurrence Relations (1 of 4: Introduction to Recurrence relations with introductory examples)

Recurrence Relations (2 of 4: Considering if the question consists of some arbitrary power of n)

Recurrence Relations (2 of 4: Considering if the question consists of some arbitrary power of n)

Recurrence Relations (3 of 4: Applying to Trigonometric Functions raised to power of n)

Recurrence Relations (3 of 4: Applying to Trigonometric Functions raised to power of n)

Recurrence Relations (4 of 4: Integration by parts with definite integrals)

Recurrence Relations (4 of 4: Integration by parts with definite integrals)

Evaluating Recurrence Relations (1 of 4: When do you apply Recurrence Relations?)

Evaluating Recurrence Relations (1 of 4: When do you apply Recurrence Relations?)

Evaluating Recurrence Relations (2 of 4: Using Integration of parts to build recurring pattern)

Evaluating Recurrence Relations (2 of 4: Using Integration of parts to build recurring pattern)

Evaluating Recurrence Relations (3 of 4: Using Recurring Pattern to group Relation in a series)

Evaluating Recurrence Relations (3 of 4: Using Recurring Pattern to group Relation in a series)

Evaluating Recurrence Relations (4 of 4: Finding a term free of integrals)

Evaluating Recurrence Relations (4 of 4: Finding a term free of integrals)

Extension 2 Exam Review (1 of 7: Integration by Substitution, Partial Fractions)

Extension 2 Exam Review (1 of 7: Integration by Substitution, Partial Fractions)

Extension 2 Exam Review (2 of 7: Recurrence Relation, Graph Transformations)

Extension 2 Exam Review (2 of 7: Recurrence Relation, Graph Transformations)

Mathematics Extension 1 Exam Review (1 of 3: Integration by substitution)

Mathematics Extension 1 Exam Review (1 of 3: Integration by substitution)

Integration by Parts (2 of 2: When the integrand doesn't look like a product)

Integration by Parts (2 of 2: When the integrand doesn't look like a product)

Integration by Parts (1 of 2: Arranging the integral with DETAIL)

Integration by Parts (1 of 2: Arranging the integral with DETAIL)

Integration with Quadratic Denominators (3 of 3: Rationalising the numerator)

Integration with Quadratic Denominators (3 of 3: Rationalising the numerator)

Integration with Quadratic Denominators (2 of 3: Distinguishing characteristics)

Integration with Quadratic Denominators (2 of 3: Distinguishing characteristics)

Integration with Quadratic Denominators (1 of 3: Introduction - why are they challenging?)

Integration with Quadratic Denominators (1 of 3: Introduction - why are they challenging?)

Partial Fractions (3 of 3: Three solution methods)

Partial Fractions (3 of 3: Three solution methods)

Partial Fractions (2 of 3: What are they?)

Partial Fractions (2 of 3: What are they?)

Partial Fractions (1 of 3: Review questions)

Partial Fractions (1 of 3: Review questions)

Integration by Unspecified Substitution (3 of 3: Trigonometric example)

Integration by Unspecified Substitution (3 of 3: Trigonometric example)

Integration by Unspecified Substitution (2 of 3: Square root example)

Integration by Unspecified Substitution (2 of 3: Square root example)

Integration by Unspecified Substitution (1 of 3: Introductory example)

Integration by Unspecified Substitution (1 of 3: Introductory example)

Intro to Further Integration (2 of 2: Foundational examples)

Intro to Further Integration (2 of 2: Foundational examples)

Intro to Further Integration (1 of 2: Expanding on prior learning)

Intro to Further Integration (1 of 2: Expanding on prior learning)

Recurrence Relations (3 of 3: Trigonometric examples)

Recurrence Relations (3 of 3: Trigonometric examples)

Recurrence Relations (2 of 3: Exponential example)

Recurrence Relations (2 of 3: Exponential example)

Recurrence Relations (1 of 3: Introduction & logarithmic example)

Recurrence Relations (1 of 3: Introduction & logarithmic example)

Recurrence Relation - worked example (1 of 2: Integration by parts)

Recurrence Relation - worked example (1 of 2: Integration by parts)

Recurrence Relation - worked example (2 of 2: Connecting the integrals)

Recurrence Relation - worked example (2 of 2: Connecting the integrals)

Logarithmic integral with recurrence relation (Exam Question 4 of 10)

Logarithmic integral with recurrence relation (Exam Question 4 of 10)