Calculus: Analytic Geometry and Calculus, with Vectors

472 Trigonometric functions

careless about quadrants in which angles lie. There is a "theory of

analytic functions" that guarantees that, in many situations, a formula

which is correct when angles lie in the first quadrant will be correct

wherever we want to use it.

When we wish to make a substitution to evaluate the integral in

(8.46) I = f

a

1

2

dx,

'--

we can clarify our work and save writing the integral by denoting it by

I, or by Il or J if we wish to distinguish it from other integrals. We

can then put x = a sin 0 and allow the variables to shift for themselves

while we write

(8.461) I=f 1 a cos 8 de = f 1 de

a2

cos,

e

x

=0+c=sin'-+ C.

a

We have evaluated an integral that previously appeared in (8.354).

The identity 1 - cos2 8 = sine 0 provides the reason why the substi-

tution x = a sin a eliminated the square roots from the integrands of the

above integrals. The identities

(8.47) sec2 0 - 1 = tan2 e, tang 0 + 1 = sect 0,

which are obtainable from the one involving sines and cosines by dividing

by cos2 9, are less familiar but are nevertheless important when we want

to use them. Their uses are illustrated below.

To evaluate the integral in

(8.471)

J -

f(x2+a2)5'x,

we look at it and generate the idea that we should try setting x = a tan 9.

Then x' = a sec2 8, so

(8.472) J = f(a2 sec2 e) a sec2 a d8 =

a2fsec

d8

=1 rcos0d8=1sin0+c=a2 J a2 a2 x2 + a2x -+ -c

the last step being assisted by a figure showing 8, a, and x in the right

way.

To evaluate the integral in

(8.473) 1

f(X2 a2),%

dx,