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Derivatives
Related Topics
Wize University Calculus 1 Textbook > Integrals
The Fundamental Theorem of Calculus - Part 1
5 Activities
Given that
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
=
0
,
\lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
=
0
,
determine
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
sin
x
\lim_{x\rightarrow0}\frac{\int_0^x\left(t^2-1\right)dt}{\sin x}
lim
x
→
0
s
i
n
x
∫
0
x
(
t
2
−
1
)
d
t
.
-1
0
1
π
\pi
π
does not exist
I don't know
Check Submission
More The Fundamental Theorem of Calculus - Part 1 Questions:
Practice: FTC I (~F2016 Final Q40)
Find
d
d
x
(
∫
x
x
2
t
2
d
t
)
\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right)
d
x
d
(
∫
x
x
2
t
2
d
t
)
.
Derivative of Integral
Find
d
d
x
∫
x
3
(
t
2
−
tan
t
)
100
d
t
\displaystyle\frac{d}{dx}\int_x^3\left(t^2-\tan t\right)^{100}dt
d
x
d
∫
x
3
(
t
2
−
tan
t
)
100
d
t
FTC I
If
g
(
x
)
=
∫
1
x
cos
t
d
t
g\left(x\right)=\int_1^x\cos t\ dt
g
(
x
)
=
∫
1
x
cos
t
d
t
, find
g
′
(
π
2
)
g'\left(\frac{\pi}{2}\right)
g
′
(
2
π
)
.
Practice: FTC I
If
g
(
x
)
=
∫
1
x
cos
t
d
t
g\left(x\right)=\int_1^x\cos t\ dt
g
(
x
)
=
∫
1
x
cos
t
d
t
, find
g
′
(
π
2
)
g'\left(\frac{\pi}{2}\right)
g
′
(
2
π
)
.
Practice: FTC I
Practice: FTC I
If
g
(
x
)
=
∫
1
x
cos
t
d
t
g\left(x\right)=\int_1^x\cos t\ dt
g
(
x
)
=
∫
1
x
cos
t
d
t
, find
g
′
(
π
2
)
g'\left(\frac{\pi}{2}\right)
g
′
(
2
π
)
.
Application of Fundamental Theorem of Calculus
For what values of
x
x
x
is the function
f
(
x
)
=
∫
x
2
2
(
t
−
1
)
d
t
f\left(x\right)=\int_{x^2}^2\left(t-1\right)dt
f
(
x
)
=
∫
x
2
2
(
t
−
1
)
d
t
decreasing?
Practice: Application of FTC I (~F2015 Final Q16)
Practice: Application of FTC I
Given that
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
=
0
,
\lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
=
0
,
determine
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
sin
x
\lim_{x\rightarrow0}\frac{\int_0^x\left(t^2-1\right)dt}{\sin x}
lim
x
→
0
s
i
n
x
∫
0
x
(
t
2
−
1
)
d
t
.
Practice: FTC I (~F2016 Final Q40)
Practice: FTC I
Find
d
d
x
(
∫
x
x
2
t
2
d
t
)
\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right)
d
x
d
(
∫
x
x
2
t
2
d
t
)
.
Practice: FTC I
Practice: FTC I
If
g
(
x
)
=
∫
1
x
cos
t
d
t
g\left(x\right)=\int_1^x\cos t\ dt
g
(
x
)
=
∫
1
x
cos
t
d
t
, find
g
′
(
π
2
)
g'\left(\frac{\pi}{2}\right)
g
′
(
2
π
)
.
simona integral ftc
If
g
(
x
)
=
∫
1
x
cos
t
d
t
\displaystyle g\left(x\right)=\int_1^x\cos t \ dt
g
(
x
)
=
∫
1
x
cos
t
d
t
, find
g
′
(
π
2
)
g'\left(\dfrac{\pi}{2}\right)
g
′
(
2
π
)
.
Evaluate Derivative of Integral
Given
f
(
x
)
=
d
d
t
∫
sin
x
cos
x
ln
t
d
t
\displaystyle f(x)=\frac{d}{dt}\int_{\sin x}^{\cos x}\ln t\ dt
f
(
x
)
=
d
t
d
∫
s
i
n
x
c
o
s
x
ln
t
d
t
, find
f
(
π
/
3
)
f(\pi/3)
f
(
π
/3
)
.
Practice: Deriv of Integral with Chain Rule
Q.
\textbf{Q.}
Q.
Find the derivative of
∫
0
ln
(
x
2
)
sin
t
2
d
t
\displaystyle \int_{0}^{\ln(x^2)}\sin{t^2}\,\text{d}t
∫
0
l
n
(
x
2
)
sin
t
2
d
t
.
Derivative of Integral
Find
d
d
x
∫
x
3
(
t
2
−
tan
t
)
100
d
t
\displaystyle\frac{d}{dx}\int_x^3\left(t^2-\tan t\right)^{100}dt
d
x
d
∫
x
3
(
t
2
−
tan
t
)
100
d
t
Practice: Application of FTC I (~F2018 Final Q40)
Practice: Application of FTC I
For what values of
x
x
x
is the function
f
(
x
)
=
∫
x
2
2
(
t
−
1
)
d
t
f\left(x\right)=\int_{x^2}^2\left(t-1\right)dt
f
(
x
)
=
∫
x
2
2
(
t
−
1
)
d
t
decreasing?
Intervals of Increase and Decrease
If the function is given by
f
(
x
)
=
∫
x
2
(
t
+
1
)
2
(
t
−
2
)
t
d
t
f(x)=\int_x^2\left(t+1\right)^2\left(t-2\right)t\ dt
f
(
x
)
=
∫
x
2
(
t
+
1
)
2
(
t
−
2
)
t
d
t
, then
f
(
x
)
f\left(x\right)
f
(
x
)
is decreasing on the interval
Applications of Integration: Position, Velocity and Acceleration
If the displacement of a particle is given by
s
(
t
)
=
∫
0
t
x
2
sin
x
d
x
s\left(t\right)=\int_0^t\frac{x^2}{\sin x}dx
s
(
t
)
=
∫
0
t
s
i
n
x
x
2
d
x
, then the acceleration at at
t
=
π
2
t=\frac{\pi}{2}
t
=
2
π
is
Find
d
d
x
(
∫
x
3
x
2
0
d
t
)
\displaystyle\frac{d}{dx}\left(\int_x^{3x^2}0\ dt\right)
d
x
d
(
∫
x
3
x
2
0
d
t
)
.
Evaluate
lim
x
→
0
∫
0
x
cos
2
t
d
t
2
x
\displaystyle\lim_{x\rightarrow0}\frac{\int_0^x\cos^2t\ dt}{2x}
x
→
0
lim
2
x
∫
0
x
cos
2
t
d
t
if it is known that
lim
x
→
0
∫
0
x
cos
2
t
d
t
=
0
\displaystyle\lim_{x\rightarrow0}\int_0^x\cos^2t\ dt=0
x
→
0
lim
∫
0
x
cos
2
t
d
t
=
0
.
Practice: FTC
Find the derivative of
∫
0
x
2
sin
t
2
d
t
\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t
∫
0
x
2
sin
t
2
d
t
.
Practice: FTC
Q.
\textbf{Q.}
Q.
Find the derivative of
∫
0
x
2
sin
t
2
d
t
\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t
∫
0
x
2
sin
t
2
d
t
.
Practice: FTC I
Find
d
d
x
(
∫
x
x
2
t
2
d
t
)
\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right)
d
x
d
(
∫
x
x
2
t
2
d
t
)
.
Find the derivative using the Fundamental Theorem of Calculus.
d
d
x
∫
x
3
csc
t
d
t
\displaystyle\frac{d}{dx}\int_x^3\csc t\ dt
d
x
d
∫
x
3
csc
t
d
t
Practice: FTC I
Find
d
d
x
(
∫
x
x
2
t
2
d
t
)
\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right)
d
x
d
(
∫
x
x
2
t
2
d
t
)
.
Practice: FTC I
If
g
(
x
)
=
∫
1
x
cos
t
d
t
g\left(x\right)=\int_1^x\cos t\ dt
g
(
x
)
=
∫
1
x
cos
t
d
t
, find
g
′
(
π
2
)
g'\left(\frac{\pi}{2}\right)
g
′
(
2
π
)
.
Derivatives
Find the derivative of
∫
−
1
2
x
3
cos
t
2
d
t
\int_{-1}^{2x^3} \cos t^2\ dt
∫
−
1
2
x
3
cos
t
2
d
t
Derivatives
Find the derivative
d
d
x
∫
0
sin
π
sin
t
2
d
t
\displaystyle \dfrac{d}{dx}\int_{0}^{\sin\pi}\sin{t^2}\,\text{d}t
d
x
d
∫
0
s
i
n
π
sin
t
2
d
t
.
Practice: FTC I
If
g
(
x
)
=
∫
x
2
e
x
cos
t
d
t
\displaystyle g\left(x\right)=\int_{x^2}^{e^x}\cos t\ dt
g
(
x
)
=
∫
x
2
e
x
cos
t
d
t
, find
g
′
(
0
)
g'\left(0\right)
g
′
(
0
)
.
Practice: Application of FTC I
Given that
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
=
0
,
\displaystyle \lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\
x
→
0
lim
∫
0
x
(
t
2
−
1
)
d
t
=
0
,
determine
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
sin
x
\displaystyle \lim_{x\rightarrow0}\frac{\displaystyle \int_0^x\left(t^2-1\right)dt}{\sin x}
x
→
0
lim
sin
x
∫
0
x
(
t
2
−
1
)
d
t
.
Find the derivative using the Fundamental Theorem of Calculus:
d
d
x
∫
sin
x
cos
x
ln
t
d
t
\displaystyle\frac{d}{dx}\int_{\sin x}^{\cos x}\ln t\ dt
d
x
d
∫
s
i
n
x
c
o
s
x
ln
t
d
t
.
Find the derivative using the Fundamental Theorem of Calculus:
d
d
x
∫
x
3
csc
t
d
t
\displaystyle\frac{d}{dx}\int_x^3\csc t\ dt
d
x
d
∫
x
3
csc
t
d
t
.
Compute the derivative
d
d
x
[
∫
x
2
1
(
e
t
+
1
)
d
t
]
\displaystyle \frac{d}{dx}\left[\int_{x^2}^{1}(e^t+1)dt\right]
d
x
d
[
∫
x
2
1
(
e
t
+
1
)
d
t
]
.
Trigonometric Functions
Let
f
(
x
)
=
1
+
∫
5
5
x
arctan
(
t
2
)
d
t
f\left(x\right)=1+\int_5^{5x}\arctan\left(t^2\right)dt
f
(
x
)
=
1
+
∫
5
5
x
arctan
(
t
2
)
d
t
Find the linearization of
f
(
x
)
f\left(x\right)
f
(
x
)
centered at
a
=
1
a=1
a
=
1
.
simona integral ftc
If
g
(
x
)
=
∫
1
x
cos
t
d
t
\displaystyle g\left(x\right)=\int_1^x\cos t \ dt
g
(
x
)
=
∫
1
x
cos
t
d
t
, find
g
′
(
π
2
)
g'\left(\dfrac{\pi}{2}\right)
g
′
(
2
π
)
.
Practice: Deriv of Integral with Chain Rule
Q.
\textbf{Q.}
Q.
Find the derivative of
∫
0
ln
(
x
2
)
sin
t
2
d
t
\displaystyle \int_{0}^{\ln(x^2)}\sin{t^2}\,\text{d}t
∫
0
l
n
(
x
2
)
sin
t
2
d
t
.
Evaluate Derivative of Integral
Given
f
(
x
)
=
d
d
t
∫
sin
x
cos
x
ln
t
d
t
\displaystyle f(x)=\frac{d}{dt}\int_{\sin x}^{\cos x}\ln t\ dt
f
(
x
)
=
d
t
d
∫
s
i
n
x
c
o
s
x
ln
t
d
t
, find
f
(
π
/
3
)
f(\pi/3)
f
(
π
/3
)
.
Derivative of Integral
Find
d
d
x
∫
x
3
(
t
2
−
tan
t
)
100
d
t
\displaystyle\frac{d}{dx}\int_x^3\left(t^2-\tan t\right)^{100}dt
d
x
d
∫
x
3
(
t
2
−
tan
t
)
100
d
t
Find the derivative using the Fundamental Theorem of Calculus:
d
d
x
∫
x
3
x
arctan
t
d
t
\displaystyle\frac{d}{dx}\int_{\sqrt{x}}^{3x}\arctan\ t\ dt
d
x
d
∫
x
3
x
arctan
t
d
t
.
Find the derivative using the Fundamental Theorem of Calculus:
d
d
x
∫
3
x
9
x
log
3
t
d
t
\displaystyle\frac{d}{dx}\int_{3^x}^{9^x}\log_3t\ dt
d
x
d
∫
3
x
9
x
lo
g
3
t
d
t
.
Find the second derivative with respect to
x
of
∫
4
6
x
t
1
+
t
d
t
\int_4^{6x}\frac{t}{1+t}dt
∫
4
6
x
1
+
t
t
d
t
Find the derivative of the integral, i.e.
d
d
x
∫
sin
x
cos
x
ln
t
d
t
\displaystyle\frac{d}{dx}\int_{\sin x}^{\cos x}\ln t\ dt
d
x
d
∫
s
i
n
x
c
o
s
x
ln
t
d
t
Find the derivative with respect to
x
x
x
of
∫
π
/
6
ln
x
(
sec
t
+
tan
t
)
d
t
\int_{\pi/6}^{\ln x}(\sec t+\tan t)dt
∫
π
/6
l
n
x
(
sec
t
+
tan
t
)
d
t
.
Find the second derivative with respect to
x
x
x
of
∫
4
6
x
t
1
+
t
d
t
\int_4^{6x}\frac{t}{1+t}dt
∫
4
6
x
1
+
t
t
d
t
.
Find
d
d
x
(
∫
x
3
x
2
0
d
t
)
\displaystyle\frac{d}{dx}\left(\int_x^{3x^2}0\ dt\right)
d
x
d
(
∫
x
3
x
2
0
d
t
)
.
Evaluate
lim
x
→
0
∫
0
x
cos
2
t
d
t
2
x
\displaystyle\lim_{x\rightarrow0}\frac{\int_0^x\cos^2t\ dt}{2x}
x
→
0
lim
2
x
∫
0
x
cos
2
t
d
t
if it is known that
lim
x
→
0
∫
0
x
cos
2
t
d
t
=
0
\displaystyle\lim_{x\rightarrow0}\int_0^x\cos^2t\ dt=0
x
→
0
lim
∫
0
x
cos
2
t
d
t
=
0
.
Applications of Integration: Position, Velocity and Acceleration
A particle's velocity in cm/s is given by the function
v
(
t
)
=
∫
1
t
2
x
d
x
v\left(t\right)=\int_1^{t^2}x\ dx
v
(
t
)
=
∫
1
t
2
x
d
x
. When is the particle's acceleration
54
54
54
cm/s
2
?
Derivatives
Find
d
d
x
[
∫
x
3
x
(
t
−
t
3
)
d
t
]
\frac{d}{dx}\left[\int_{x^3}^{\sqrt{x}}\left(\sqrt{t}-t^3\right)\ dt\right]
d
x
d
[
∫
x
3
x
(
t
−
t
3
)
d
t
]
.
Application of Fundamental Theorem of Calculus
For what values of
x
x
x
is the function
f
(
x
)
=
∫
x
2
2
(
t
−
1
)
d
t
f\left(x\right)=\int_{x^2}^2\left(t-1\right)dt
f
(
x
)
=
∫
x
2
2
(
t
−
1
)
d
t
decreasing?
Practice: Application of FTC I (~F2015 Final Q16)
Practice: Application of FTC I
Given that
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
=
0
,
\lim_{x\rightarrow0}\int_0^x\left(t^2-1\right)dt=0,\
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
=
0
,
determine
lim
x
→
0
∫
0
x
(
t
2
−
1
)
d
t
sin
x
\lim_{x\rightarrow0}\frac{\int_0^x\left(t^2-1\right)dt}{\sin x}
lim
x
→
0
s
i
n
x
∫
0
x
(
t
2
−
1
)
d
t
.
Practice: FTC I (~F2016 Final Q40)
Practice: FTC I
Find
d
d
x
(
∫
x
x
2
t
2
d
t
)
\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right)
d
x
d
(
∫
x
x
2
t
2
d
t
)
.
FTC I
If
g
(
x
)
=
∫
1
x
cos
t
d
t
g\left(x\right)=\int_1^x\cos t\ dt
g
(
x
)
=
∫
1
x
cos
t
d
t
, find
g
′
(
π
2
)
g'\left(\frac{\pi}{2}\right)
g
′
(
2
π
)
.
Practice: FTC I
Find
d
d
x
(
∫
x
x
2
t
2
d
t
)
\frac{d}{dx}\left(\int_x^{x^2}t^2dt\right)
d
x
d
(
∫
x
x
2
t
2
d
t
)
.
Practice: FTC I
If
g
(
x
)
=
∫
1
x
cos
t
d
t
g\left(x\right)=\int_1^x\cos t\ dt
g
(
x
)
=
∫
1
x
cos
t
d
t
, find
g
′
(
π
2
)
g'\left(\frac{\pi}{2}\right)
g
′
(
2
π
)
.
Practice: FTC
Q.
\textbf{Q.}
Q.
Find the derivative of
∫
0
x
2
sin
t
2
d
t
\displaystyle \int_{0}^{x^2}\sin{t^2}\,\text{d}t
∫
0
x
2
sin
t
2
d
t
.