Part II: The Answers
356
Answers
501–600
To use the fundamental theorem of calculus, you need to have the variable in
the upper limit of integration. Therefore, to find the derivative of the function
f x
e dt
t
x
( ) ln
=
+
( )
∫ 2 1
4
, flip the limits of integration and change the sign of the integral:
f x
e dt
e dt
t
x
t
x
( ) ln
ln
=
= −
+
( )
+
( )
∫
∫
2
2
1
4
4
1
Note also that to find d
dx
g t dt
a
h x
( )
( )
∫
, you can use the substitution u = h(x) and then
apply the chain rule as follows:
d
dx
g t dt
d
dx
g t dt
d
du
g t dt du
dx
g u du
d
a
h x
a
u
a
u
( )
( )
( )
( )
( )
∫
∫
∫
=
=
(
)
= ′
(
) x x
g h x
du
dx
= ′ (
)
(
)
( )
All this tells you to substitute the upper limit of integration into the integrand and multiply by the derivative of the upper limit of integration. Therefore, the derivative of
f x
e dt
t
x
( )
ln
= −
+
( )
∫ 4
1
2
is
′
= −
(
) +




= −
+
+




= −
+
( )
f x
e
x
x
x
x
x
x
x
( )
( )
(
)
ln
2 1
2
2
2
1
1
2
1
2
1
2
552.
sin cos
sin cos
x
x
x
2
+
(
)
(
)
Part of the fundamental theorem of calculus states that if the function g is continuous
on [a, b], then the function f defined by
f x
g t dt
a x b
a
x
( )
( )
(
)
=
≤ ≤
∫
where
is continuous on [a, b] and is differentiable on (a, b). Furthermore, f '(x) = g(x).
To use the fundamental theorem of calculus, you need to have the variable in
the upper limit of integration. Therefore, to find the derivative of the function
f x
t
t dt
x
( )
sin
cos
=
+
(
)
∫
2
1
, first flip the limits of integration and change the sign:
f x
t
t dt
t
t dt
x
x
( )
sin
sin
cos
cos
=
+
(
)
= −
+
(
)
∫
∫
2
1
2
1
Note also that to find d
dx
g t dt
a
h x
( )
( )
∫
, you can use the substitution