7. Integrals

What you will be able to do

• State the question integral calculus is set up to answer, in terms of a given derivative and an unknown function
• Read any row of the chapter's two-column list in both directions and say which direction is the definition and which is the consequence
• List the five properties the section states and say which two are used in practice and which two are used to prove them
• State the change-of-variable formula and identify each of its three pieces in a worked example
• Say why a product of two ratios with different arguments defeats substitution, and what has to happen first
• Recite the six formulas as three related pairs and say what distinguishes each pair
• Decide whether a given rational function is proper or improper, using the chapter's own criterion
• Derive the by-parts formula by integrating the product rule, and identify each term in the result
• State the exponential result and identify, in a given integrand, the function and its derivative that make it apply
• Explain what the two attached numbers do to an indefinite integral, in terms of families and single values
• Describe the region the chapter identifies with a definite integral, naming all four of its boundaries
• State the chapter's four-step procedure and identify which step the limits enter at

What this chapter assumes you already know

• The derivative of a function of one real variable, from Chapter 5 of Part I
• The derivatives of the six trigonometric ratios, the exponential, the natural logarithm and a general power, all as a memorised list
• What it means for a derivative to vanish everywhere on an interval
• Interval notation, and the phrase for every point of an interval
• Set-builder notation, and reading a set whose members are indexed by a parameter
• The two inverse trigonometric functions whose derivatives appear in the standard list, from Chapter 2 of Part I
• The derivative of a general power, of the six trigonometric ratios, of the exponential and of the natural logarithm, from Chapter 5 of Part I
• The derivatives of the inverse sine, inverse cosine and inverse tangent, from Chapter 5 of Part I

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

• "The constant is a formality you tack on at the end."
• "Every anti derivative of a function differs from a given one by a constant — that's obvious."
• "Integration is just differentiation done backwards, so the same rules apply in reverse."
• "The exponential of a negative square has no anti derivative."
• "Adding a constant gives a different function, so the answer is ambiguous and therefore useless."
• "An anti derivative and an integral are two different things."
• "The letter under the d carries meaning, so changing it changes the answer."
• "The portrait means he invented this."