Calculus AB and Calculus BC

CHAPTER 5 Antidifferentiation

B. BASIC FORMULAS

Familiarity with the following fundamental integration formulas is essential.

Image

Image

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

Image

EXAMPLE 2

Image

EXAMPLE 3

Image

EXAMPLE 4

Image is integrated most efficiently by using formula (3) with u = 1 − 3x and du = u (x)dx = −3 dx.

Image

EXAMPLE 5

Image where u = 2x3 − 1 and du = u (x) dx = 6x2 dx; this, by formula (3), equals

Image

EXAMPLE 6

Image du, where u = 1 − x and du = −1 dx; this, by formula (3) yields Image

EXAMPLE 7

Image

EXAMPLE 8

Image

EXAMPLE 9

Image

EXAMPLE 10

Image

EXAMPLE 11

Image

EXAMPLE 12

Image

EXAMPLE 13

Image

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

Image

EXAMPLE 15

Image

EXAMPLE 16

Image 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

Image

EXAMPLE 18

Image (by long division) = −x − ln |1 − x| + C.

EXAMPLE 19

Image

EXAMPLE 20

Image

EXAMPLE 21

Image

EXAMPLE 22

Image

EXAMPLE 23

Image

EXAMPLE 24

Image

EXAMPLE 25

Image + C by (3) with u = tan t and du = u (t) dt = sec2 t dt.

EXAMPLE 26

Image

EXAMPLE 27

Image by (4)

with u = 1 + 2Image and Image

EXAMPLE 28

Image

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

EXAMPLE 29

Image

EXAMPLE 30

Image using the trigonometric identity Image

EXAMPLE 31

Image

EXAMPLE 32

Image

EXAMPLE 33

Image

EXAMPLE 34

Image

EXAMPLE 35

Image

EXAMPLE 36

Image

EXAMPLE 37

Image

EXAMPLE 38

Image

EXAMPLE 39

Image

EXAMPLE 40

Image

EXAMPLE 41

Image

BC ONLY