8.4 Integration by trigonometric andother substitutions 475
Problems 8.49
1 By making an appropriate figure and trigonometric substitution,fill in all
of the missing details which show that
(a) r x21-a2dx = r sec 9 dB = log (x -f x2 - a2) + c
(b) !(a2 x2dx = a f sec 8 d9
=flog a+x +c logo+x+c
a a2-,x2 2a a-x (JxI <a)
(c) f
x2 a2
dx =
Q
f csc 8 dB
(^1) l x+ a (^1) x- a
- C
a
log
x2 --a 2
+ 2alegx + a
(d) f (a2 - x2)3 dx = a° f cos' 0 dO
2 When x > 0, the identity
(1)
1 + 1 = z2^1 - x2 + 1
x x x 1 + X2
shows that
(2) f 1 +^1 dx = f (1 +
x x 1+x2
(x > a)
The first integral on the right is easily evaluated, and the second can be simplified
by a trigonometric substitution. Use these ideas to derive the formula
(3) f rl ± .2dx = 1 x2 + log x - log (1 + 1 -+x2) + c.
3 Sketch graphs of y = ex and y = log x in a single figure and note that each
is the mirror image of the other in the line y = x. Observe that the arc C, on
the graph of y = ex which joins two points (p,,q,) and (p2,Q2) on the graph of
y = ex is congruent to the arc C2 on the graph of y = log x which joins the two
points (q,,p,) and (g2,p2) on the graph of y = log x. LettingL denote the length
of C, and C2, show that
(1)
L =
f
p:
1 e2x dx = f 'D'V+1 dx.
P, 'D, x2
This gives the formula
(2)
J
1 + eit dt =
I.
`y' 1 + 1 dx.
P
1
i en2 x2
To show that errors and misprints have not led us to an incorrect formula, show
that the substitution i = log x converts the first integral in (2) into the second
integral in (2).