﻿ ﻿BASIC FORMULAS - Antidifferentiation - Calculus AB and Calculus BC

## CHAPTER 5 Antidifferentiation

### B. BASIC FORMULAS

Familiarity with the following fundamental integration formulas is essential.  All the references in the following set of examples are to the preceding basic formulas. In all of these, whenever u is a function of x, we define du to be u (x) dx; when u is a function of t, we define du to be u (t) dt; and so on.

EXAMPLE 1 EXAMPLE 2 EXAMPLE 3 EXAMPLE 4 is integrated most efficiently by using formula (3) with u = 1 − 3x and du = u (x)dx = −3 dx. EXAMPLE 5 where u = 2x3 − 1 and du = u (x) dx = 6x2 dx; this, by formula (3), equals EXAMPLE 6 du, where u = 1 − x and du = −1 dx; this, by formula (3) yields EXAMPLE 7 EXAMPLE 8 EXAMPLE 9 EXAMPLE 10 EXAMPLE 11 EXAMPLE 12 EXAMPLE 13 If the degree of the numerator of a rational function is not less than that of the denominator, divide until a remainder of lower degree is obtained.

EXAMPLE 14 EXAMPLE 15 EXAMPLE 16 with u = 5 + 2 sin x. The absolute-value sign is not necessary here since (5 + 2 sin x) > 0 for all x.

EXAMPLE 17 EXAMPLE 18 (by long division) = −x − ln |1 − x| + C.

EXAMPLE 19 EXAMPLE 20 EXAMPLE 21 EXAMPLE 22 EXAMPLE 23 EXAMPLE 24 EXAMPLE 25 + C by (3) with u = tan t and du = u (t) dt = sec2 t dt.

EXAMPLE 26 EXAMPLE 27 by (4)

with u = 1 + 2 and EXAMPLE 28 with u = cos x; cos 2x + C by (6), where we use the trigonometric identity sin 2x = 2 sin x cos x.

EXAMPLE 29 EXAMPLE 30 using the trigonometric identity EXAMPLE 31 EXAMPLE 32 EXAMPLE 33 EXAMPLE 34 EXAMPLE 35 EXAMPLE 36 EXAMPLE 37 EXAMPLE 38 EXAMPLE 39 EXAMPLE 40 EXAMPLE 41 BC ONLY

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