Page 104 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 5] INTEGRALS 95

If the open interval on which f is continuous includes a and b, then we may write
b
ð
f ðxÞ dx ¼ FðbÞ FðaÞ: (See Problem 5.11)
a
This is the usual form in which the theorem is used.

ð 2 x 3 ð 2
2
2
2
EXAMPLE. To evaluate x dx we observe that F ðxÞ¼ x , FðxÞ¼ þ c and x dx ¼ 2 3 3 þ c
0
1 3 1
2 3 1 3
7
1 3 þ c ¼ . Since c subtracts out of this evaluation it is convenient to exclude it and simply write .
3 3 3 3
GENERALIZATION OF THE LIMITS OF INTEGRATION
The upper and lower limits of integration may be variables. For example:
"# cos x
t 2 2
ð cos x 2
¼ðcos x sin xÞ=2
tdt ¼
sin x 2
sin x
In general, if F ðxÞ¼ f ðxÞ then
0
ð
vðxÞ
f ðtÞ dt ¼ F½vðxފ ¼ F½uðxފ
uðxÞ

CHANGE OF VARIABLE OF INTEGRATION
Ð
If a determination of f ðxÞ dx is not immediately obvious in terms of elementary functions, useful
results may be obtained by changing the variable from x to t according to the transformation x ¼ gðtÞ.
(This change of integrand that follows is suggested by the differential relation dx ¼ g ðtÞ dt.) The funda-
0
mental theorem enabling us to do this is summarized in the statement
ð ð
f fgðtÞgg ðtÞ dt
0
f ðxÞ dx ¼ ð6Þ
where after obtaining the indefinite integral on the right we replace t by its value in terms of x, i.e.,
1
t ¼ g ðxÞ. This result is analogous to the chain rule for differentiation (see Page 69).
The corresponding theorem for definite integrals is
ð b ð
f fgðtÞgg ðtÞ dt
0
f ðxÞ dx ¼ ð7Þ
a
1
1
where gð Þ¼ a and gð Þ¼ b, i.e., ¼ g ðaÞ, ¼ g ðbÞ. This result is certainly valid if f ðxÞ is con-
tinuous in ½a; bŠ and if gðtÞ is continuous and has a continuous derivative in @ t @ .

INTEGRALS OF ELEMENTARY FUNCTIONS
The following results can be demonstrated by differentiating both sides to produce an identity. In
each case an arbitrary constant c (which has been omitted here) should be added.