355
Answers
501–600
Answers and Explanations
550.
–cos
3 
x
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 dt
x
( )
(
)
sin
=
−
∫ 1
2
0
, first flip the limits of integration and change the sign:
f x
t dt
t dt
x
x
( )
(
)
(
)
sin
sin
=
−
= −
−
∫
∫
1
1
2
0
2
0
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
t dt
x
( )
(
)
sin
= −
−
∫ 1
2
0
is
′
= − −
(
)
= −
−
= −
= −
f x
x
x
x
x
x
x
( )
(sin ) (cos )
cos ( sin )
cos (cos )
cos
1
1
2
2
2
3 x x
Note that the identity 1 – sin
2 
x = cos
2
x was used to simplify the derivative.
551.
–2x
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).