Unit 2 of 7
Recognizing symbolic methods
- 6 chapters
- Chapters 5 to 10
Each chapter shows how to recognize one symbolic method, from substitution and integration by parts to a half-angle identity, trigonometric substitution, partial fractions and completing the square.
Substitution looks for an inner function whose derivative is also present and changes the bounds with the variable. Integration by parts reverses the product rule and trades a product for a boundary term and a simpler integral. A double-angle identity rewrites sine squared as a constant minus an oscillation that integrates to zero. A sine substitution turns the square root of 4 minus x squared into a cosine, partial fractions split a factorable denominator into logarithms, and completing the square leads an unfactorable quadratic to an arctangent. Each answer is checked by differentiating.
Every chapter will be part of one long video. The video is coming soon, and each chapter page already has the definitions and an example for every key term.
Substitution, integration by parts and trigonometric identities
Substitution for a hidden inner variable
How u-substitution runs the chain rule in reverse, turning an inner function and its derivative into a new variable with the differential du and changed bounds, and when the substitution must be one-to-one.
Chapter 5
Integration by parts for a useful product
How integration by parts runs the product rule in reverse, trading the integral of a product for a boundary term and a simpler integral, how to choose u and dv, and how the result reads as a complementary area.
Chapter 6
Use a half-angle identity for a squared sine
How the power-reducing form of the half-angle identity rewrites sine squared with the cosine of a doubled angle, and why the integral of sine squared from 0 to pi is pi over 2 by cancellation over a full period.
Chapter 7
Trigonometric substitution, partial fractions and completing the square
Trigonometric substitution for a circle inside a square root
How trigonometric substitution with a sine turns the square root of 4 minus x squared into a cosine, using a circle parameterization, angle bounds, the nonnegative square root and the inverse sine.
Chapter 8
Partial fractions for a factorable denominator
How partial fraction decomposition splits a rational function with a factorable denominator into simple fractions, found by comparing coefficients and integrated as logarithms when the poles lie outside the interval.
Chapter 9
Completing the square for a shifted quadratic
How completing the square writes an irreducible quadratic with negative discriminant as a shifted and scaled copy of 1 plus u squared, so that a shift and scale substitution leads to the inverse tangent antiderivative.
Chapter 10