Substitution, Parts and Partial Fractions
Substitution reverses the chain rule, integration by parts reverses the product rule, and partial fractions decompose rational integrands.
What is Substitution, Parts and Partial Fractions?
Substitution reverses the chain rule, integration by parts reverses the product rule, and partial fractions decompose rational integrands.
Key formula / rule: Substitution requires the derivative of the inner function to be present up to a constant.
Key points
- Apply substitution to reduce an integral to a standard form.
- Compute integrals of rational functions using partial fractions.
Common exam trap
Forgetting to change the limits after substituting in a definite integral.
Definitions
- Term
Substitution, Parts and Partial Fractions
- Meaning
Substitution reverses the chain rule, integration by parts reverses the product rule, and partial fractions decompose rational integrands.
- Term
Substitution, Parts and Partial Fractions — explanation
- Meaning
Each technique targets a structural feature of the integrand: an inner function, a product of unlike functions, or a factorable denominator. Choosing the right one is the entire difficulty.
Learning objectives
Apply substitution to reduce an integral to a standard form.
Compute integrals of rational functions using partial fractions.
Formulae
- Key point
Substitution requires the derivative of the inner function to be present up to a constant.
- Key point
For parts, choose u by the ILATE ordering.
- Key point
Partial fractions require the numerator ° to be lower than the denominator °.
Prerequisites
MAT-U2-IC-T1-S1-C1
Common mistakes
Forgetting to change the limits after substituting in a definite integral.
Applying partial fractions without first dividing an improper rational function.
Keywords
Substitution
Parts
Partial
Fractions
Practice preview
Evaluate the integral: ∫ x cos x dx…
easy
Evaluate the integral: ∫ e^x (tan x + sec^2 x) dx…
medium
Evaluate the integral: ∫ 1 / (sqrt(x) * (1 + x)) dx…
hard