APPENDIXE CALCULUSREVIEW 480
E.3 Integrals
Letf be a function of the variablex. Imagine the range ofxbetween the limitsx^0 andx^00
to be divided into many small increments of sizeÅxi.iD1; 2; : : :/. Letfibe the value of
fwhenxis in the middle of the range of theith increment. Then theintegral
Zx 00
x^0
fdx
is the sum
P
ifiÅxiin the limit as eachÅxiapproaches zero and the number of terms in
the sum approaches infinity. The integral is also the area under a curve off plotted as a
function ofx, measured fromxDx^0 toxDx^00. The functionf is theintegrand, which
is integrated over the integration variablex.
This book uses the following integrals:
Zx 00
x^0
dxDx^00 x^0
Zx 00
x^0
dx
x
Dln
x^00
x^0
Zx 00
x^0
xadxD
1
aC 1
.x^00 /aC^1 .x^0 /aC^1
(ais a constant other than 1 )
Zx 00
x^0
dx
axCb
D
1
a
ln
ax^00 Cb
ax^0 Cb
(ais a constant)
Here are examples of the use of the expression for the third integral withaset equal to 1
and to 2 :
Zx 00
x^0
xdxD
1
2
.x^00 /^2 .x^0 /^2
Zx 00
x^0
dx
x^2
D
1
x^00