Calculus 5.4 Fundamental Theorem of Calculus (problems inspired by Stewart Calculus © 1997, 1998)
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the given function.
1)
€
f(x)=tsin t
0
x
∫dt
€
"
f (x)=xsin x
2)
€
g(x)=t3−4
( )
2
6
x
∫dt
€
"
g (x)=x3−4
( )
2
3)
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h(u)=1
3+t5
π
u
∫dt
€
"
h (u)=1
3+u5
4)
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f(x)=sin t4
x
−3
∫dt
€
"
f (x)=−sin x4
5)
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f(x)=tsin t
0
x3
∫dt
€
"
f (x)=3x2x3sin x3
( )
6)
€
g(x)=k3
k2−3
1
x
∫dk
€
"
g (x)=
x
( )
3
x−3
$
%
&
&
&
'
(
)
)
)
1
2x
$
%
&
'
(
) =x
2x−6
7)
€
h(t)=cos x2
0
1/ t
∫dx
€
"
h (t)=cos 1
t
#
$
%
&
'
(
2
−1
t2
#
$
%
&
'
(
8)
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f(x)=tcos t2
( )
π
sin x
∫dt
€
"
f (x)=sin xcos sin x
( )
2
#
$
% &
'
( cos x
( )
9)
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f(x)=y−2
y+3
3x
x
∫dy =y−2
y+3
3x
0
∫+y−2
y+3
0
x
∫
!
f(x)=−33x−2
3x+3
#
$
%&
'
(+x−2
x+3
10)
€
g(x)=1
3−t2
cos x
x3
∫dt =1
3−t2
cos x
0
∫+1
3−t2
0
x3
∫
€
"
g (x)=sin x
3−(cos x)2+3x2
3−x3
( )
2
