Calculus: Analytic Geometry and Calculus, with Vectors

8.4 Integration by trigonometric and other substitutions 471

The required conclusion (8.441) then follows from the fact that a and #
are chosen such that u(a) = a and u(/3) = b. Careful attention must

be given to the manner in which the new limits of integration are deter-
mined. When we set x = u(t), we must determine a such that x is a
when t is a and we must determine 0 such that x is b when t = g.
We are now ready to attack (8.41) and, since this is the first time we
have made a substitution (or changed the variable) in an integral, we

proceed with great caution. Supposing that a > 0 and that -a 5x 5

a, we set

(8.45) F(x) = f a2 --x 2 dx

and let x = a sin 0, where -7r/2 :_50 < a/2. Figure 8.451 always helps

x

-Q--.z

Figure 8.451

us to see what we are doing on occasions like this. Since dx/do = a cos 0,

we obtain

(8.452) F(a sin 0) = f a2 -cos' 0 a cos o do.

But cos 0 > 0 when -Ir/2 5 0 5 7r/2, so

r 2

(8.453) F(a sin 0) = a2

J

cos2 0 do = 2

f

(1 + cos 20) do

a

=

2

[o+ sin 20] + c

_ 4[a2o + a2 sin 0 cos o] + c.

Since each x in the range -1 < x 5 1 is obtained for a 0 in the interval

-a/2 < 0 < 7r/2, we can use Figure 8.451 and the fact that cos 0 > 0

to return to the original variable x so that

(8.454) F(x) [assin-' + x a2 - x2J + C.

a

This gives the formula

[ l

(8.455) = 4 LagSir ,71

a

x -I-x a2 - x2J -}- c.

We are not always so careful about all of the details, and we do not always

get correct answers either. There is a reason why we can sometimes be