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CHAPTER 24
24.14
24.15
24.16
24.17
24.18
24.19
24.20
24.21
24.22
24.23
24.24
24.25
24.26
24.27
In Problems 24.17-24.29, evaluate the given antiderivative.
Choose a = 3
2 = 9 in Problem 24.21:
by Problem 24.23.
This is a special case of the general law
for any
constant
Then, noting that u > 0,
•n*.
D,(ir') = D s (e''"*) = e'
lnir • D,(x In IT) = e"
ln " • In TT = In ir • TT*.
In e
2 '.
D,(lne
2 *) = D,(2*) = 2.
e
A - e".
D f (e* - e~') = D x (e") - D x (e'") = e" - (-e'
x ) = e" + e'\ Here, D x (e~*) = -e~" is taken from Problem 24.7.
J e
3 ' dx.
Let u = 3x, du = 3dx. Then j e
3 " dx = ! > $ e" du = %e" + C = ^e** + C.
J e-' Let « = -*, du = -dx. Then / e"* rfx= -J e" rfw = -e" + C= -e "' + C.
$e*Ve^2dx
Let u = e'-2, du = e'dx. Then J e'^e* -2dx = f «"
2 du = §w
3 '
2 + C= i(\V -2)
3 + C.
fe
cosi sinA:djc.
J e
cos ' sin x dx = -e
cos * + C, by Problem 24.9.
Ja'dx, for a^l.
a* = <•*
ln °. So, let M = (In a)^:, dw = (In a) dr. Then
S3
2 *dx,
J Let M = ax, du = a dx. Then
/V?djc.
/Ar"djc.
J e'e
2
" dx.
J eV +2x dx = Se
3 'dx= ^e
3 ' + C.
Let a = e* + l, du = e'dx.
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