Calculus: Analytic Geometry and Calculus, with Vectors

474 t Trigonometric functions

It is always a good idea to display u and v' in one lineand u' and v in a
lower line and to know that the formula (8.482) for integrationby parts
says that the integral of the product ofthe top two is equal to the product
of u and v minus the integral of the product of the bottomtwo.f Thus

(8.484) Jz = a2 sec 0 tan 0 - a2f sec' 9 do.

The last integral seems to be quite as recalcitrant asJ2, and this really

is true because

a2f sec' 0 dO = a2f sect 8 sec 0 dO = a2f [1 + tan2 0] sec 0 d9

= a2f sec o do + a2f tan2 0 sec 0d8

= a2 log (sec o + tan 0) + c + J2

Instead of developing faint hearts by throwing everything into a waste-

basket, we substitute our last result into (8.484) and get

(8.485) J2 = a2 sec titan o - a2log (sec o + tan o) - c - J2

If we can now suddenly remember that we are trying tofind J2, we can

transpose the term -J2 (or add J2 toboth sides of our equation if trans-

posing is onerous) and divide by 2 to obtain

(8.486) J2 = .5a2 sec a tan 0 - ..a2 log (sec 0 + tan 0) + c,

where the new constant c, like the old -c/2, can be any constant. With

the aid of a figure showing how 0, a, and x are related, we obtain

/'

(8.487) J2= J x2-VT x2-a2

1

X -I- x2 - a2

-a2 log + c.

a

If we wish, we can write the logarithm of the quotient as a difference of

logarithms and combine the constant part with the c to obtain the formula

(8.488) J2 = f x2 - a2 dx = jx x2 - a2

• ka2 log (x + V'rx2- a2) + c,

in which c has another value. The answers in (8.487) and (8.488) look

different, but they are both correct. The last one may seem simpler,

but the first one gives c = 0 if J2 = 0 when x = a.

We should recognize the fact that all or nearly all of the integration

formulas in the text and problems of this section appear in books of

tables. We should hear more than once that books of tables are often

used as labor-saving devices, but that many persons like to derive the

formulas they use to preserve and improve their mathematical acumen.

t These matters are very important, and it is required that we think about them more

than once. Our concentrated attack upon integration by parts appears in Section 9.5.