Calculus: Analytic Geometry and Calculus, with Vectors

514 Exponential and logarithmic functions

Remark: We should not be too busy to see how this formula leads to another
that also appears in tables of integrals. Let

(2) X=ax2+bx+c,

where a, b, c are constants for which a > 0 and b2 - 4ac > 0. Observe that

(3) X = a (x - p) (x - q),

where p and q are the values of x for which X = 0, so that

-b - yba cs 1 bc

(4) p= 2a q 2a p-q -" a

Substituting in (1) gives the formula

5 1 dx=^1 log

r tax + b - b2 - 4ac +c.

() J X b2-4ac l2ax+b+ 02-4ac

We should know that additional tricks produce additional formulas that appear

in tables of integrals. When a > 0, integrating the first and last members of

the identity

(6) x

1 (2ax+b) -b_ 1 tax+b b 1

ax2+bx+c 2a ax2+bx+c 2aax2+bx+c 2aax2+bx+c

gives the integral formula

(7) I X dx dx.

2a

log 1XI 2 ffX

When X is defined by (2) and b2 - 4ac s 0 and n 0 1, differentiation and sim-

plification give

(8)

d tax + b= -2a(2n - 3)-(n - 1)(b2 - 4ac)

dx Xa-1 X.-1 Xn

and hence

r 1 2ax + b _ 2a(2n - 3).^1

(9) J Xn

dx

= - (n- 1)(b2 - 4ac)Xn-1 (n - 1)(b22 - 4ac) J

X; ;=i, dx.

The formulas (7) and (9) are examples of reduction formulas that sometimes

enable us to express given integrals in terms of other integrals that are more

easily evaluated. Persons who like everything in mathematics can easily become

interested in (9) and similar formulas that appear in books of tables. But the

formulas are rarely used (most people never use them), and with only a twinge

of regret we decline to invest our good time in consideration of examples more or

less like f (x2 - 5x - 1)-' dx.