Calculus: Analytic Geometry and Calculus, with Vectors

4.1 Indefinite integrals 205

The following little table gives two versions
and most useful integration formulas.

xn+i

(4.171)

/

xn dx = n +

(4.172) J sin x dx = -cos x+c

(4.173) fcosxdx-_sinx+c

(4.174)

J

ex dx = ex + c

(4.175) J x A= log Jxl + c

of each of the five simplest

I

ndu _ un+1

u

dx

dx

n

1 +

du

c

J sin u dx = - cos u

ax

J cos u dx dx =

+ c

sinu+c

J eu

du

dx = + c

I du

J

udxdx=logJul +c

In the formulas of the second column, u is supposed to be a differentiable
function of x. Subject to the requirement that n 0 -1 in (4.171), and
that x and u are confined to intervals over which the integrands in (4.171)
and (4.175) exist, these formulas are proved by observing that they have
the form (4.14) whereF'(x) = f(x). We need not learn all of the formulas

we see, but the formulas in the above table are used so often that they
must be learned.
When the formulas in the column on the right are being used, presence

of the factor du/dx must be carefully observed. It is not correct to

think of u as being sin x and to claim that use of (4.171) shows that the

members of the formula

(4.181) sine x dx 76

sing x

3 + c

are equal. We can, however, think of u as being sin x and read the, left

member of the formula

(4.182) f sin' x cos x dx =

sin' x

+ c

I

in the form "integral of u to the nth power dee u dee x dee x" and then

apply (4.171) to obtain the right member.

It is not correct to claim that the members of the formula

(4.183)

J

(5x + 7)2 dx 54(5x 3

7)s

+ c

are equal. We can, however, let I denote the left member, observe that

the integrand has the form un, where du/dx = 5, and write

(4.184) 1 = 5

J

(5x + 7)2(5) dx =5 (5x

3

7)1+ c.