Resources
Testimonials
Plans
Sign in
Sign up
Resources
Testimonials
Plans
Home
Math Problems
Calculus
Evaluate definite integrals using the power rule
Calculate the integral and write your answer in simplest form.
\newline
∫
5
x
5
4
6
d
x
\int \frac{5 \sqrt[4]{x^{5}}}{6} \mathrm{dx}
∫
6
5
4
x
5
dx
\newline
Answer:
Get tutor help
Question
\newline
Evaluate
∫
x
2
+
1
x
2
−
3
d
x
\int \frac{x^{2}+1}{x^{2}-3} \, dx
∫
x
2
−
3
x
2
+
1
d
x
\newline
Provide your answer below:
Get tutor help
∫
1
3
x
+
12
d
x
=
\int \frac{1}{3x+12} \, dx =
∫
3
x
+
12
1
d
x
=
Get tutor help
Let
f
f
f
be the function defined by
f
(
x
)
=
6
x
f(x)=6 \sqrt{x}
f
(
x
)
=
6
x
. If three subintervals of equal length are used, what is the value of the right Riemann sum approximation for
∫
3
7.5
6
x
d
x
\int_{3}^{7.5} 6 \sqrt{x} d x
∫
3
7.5
6
x
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
Get tutor help
Let
f
f
f
be the function defined by
f
(
x
)
=
2
x
f(x)=\frac{2}{x}
f
(
x
)
=
x
2
. If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
3
9
2
x
d
x
\int_{3}^{9} \frac{2}{x} d x
∫
3
9
x
2
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
Get tutor help
Let
f
f
f
be the function defined by
f
(
x
)
=
5
x
f(x)=\frac{5}{x}
f
(
x
)
=
x
5
. If five subintervals of equal length are used, what is the value of the right Riemann sum approximation for
∫
2
3
5
x
d
x
\int_{2}^{3} \frac{5}{x} d x
∫
2
3
x
5
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
Get tutor help
Let
f
f
f
be the function defined by
f
(
x
)
=
3
x
f(x)=3^{x}
f
(
x
)
=
3
x
. If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
0
3
3
x
d
x
\int_{0}^{3} 3^{x} d x
∫
0
3
3
x
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
Get tutor help
Let
f
f
f
be the function defined by
f
(
x
)
=
5
x
f(x)=5^{x}
f
(
x
)
=
5
x
. If six subintervals of equal length are used, what is the value of the right Riemann sum approximation for
∫
0
1.5
5
x
d
x
\int_{0}^{1.5} 5^{x} d x
∫
0
1.5
5
x
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
Get tutor help
Let
f
f
f
be the function defined by
f
(
x
)
=
6
x
f(x)=\frac{6}{x}
f
(
x
)
=
x
6
. If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
1
8.5
6
x
d
x
\int_{1}^{8.5} \frac{6}{x} d x
∫
1
8.5
x
6
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
Get tutor help
Let
f
f
f
be the function defined by
f
(
x
)
=
2
x
f(x)=\frac{2}{x}
f
(
x
)
=
x
2
. If five subintervals of equal length are used, what is the value of the right Riemann sum approximation for
∫
1
6
2
x
d
x
\int_{1}^{6} \frac{2}{x} d x
∫
1
6
x
2
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
Get tutor help
Let
f
f
f
be the function defined by
f
(
x
)
=
x
3
f(x)=x^{3}
f
(
x
)
=
x
3
. If six subintervals of equal length are used, what is the value of the midpoint Riemann sum approximation for
∫
0
3
x
3
d
x
\int_{0}^{3} x^{3} d x
∫
0
3
x
3
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
Get tutor help
Find the value of
∫
4
8
10
d
x
7
−
2
x
\int_{4}^{8} \frac{10 d x}{7-2 x}
∫
4
8
7
−
2
x
10
d
x
. Express your answer as a constant times
ln
3
\ln 3
ln
3
.
\newline
Answer:
□
ln
3
\square \ln 3
□
ln
3
Get tutor help
Evaluate
∫
0
16
(
5
e
−
0.25
x
−
3
)
d
x
\int_{0}^{16}\left(5 e^{-0.25 x}-3\right) d x
∫
0
16
(
5
e
−
0.25
x
−
3
)
d
x
and express the answer in simplest form.
\newline
Answer:
Get tutor help
Evaluate the integral and express your answer in simplest form.
\newline
∫
3
1
+
9
x
2
d
x
\int \frac{3}{1+9 x^{2}} d x
∫
1
+
9
x
2
3
d
x
\newline
Answer:
Get tutor help
Evaluate
∫
0
2
(
6
e
−
0.5
x
−
4
x
)
d
x
\int_{0}^{2}\left(6 e^{-0.5 x}-4 x\right) d x
∫
0
2
(
6
e
−
0.5
x
−
4
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
Get tutor help
Evaluate
∫
0
4
(
5
e
−
0.5
x
−
2
x
)
d
x
\int_{0}^{4}\left(5 e^{-0.5 x}-2 x\right) d x
∫
0
4
(
5
e
−
0.5
x
−
2
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
Get tutor help
Evaluate
∫
0
6
(
7
e
0.5
x
−
5
)
d
x
\int_{0}^{6}\left(7 e^{0.5 x}-5\right) d x
∫
0
6
(
7
e
0.5
x
−
5
)
d
x
and express the answer in simplest form.
\newline
Answer:
Get tutor help
Evaluate the integral and express your answer in simplest form.
\newline
∫
4
16
+
25
x
2
d
x
\int \frac{4}{16+25 x^{2}} d x
∫
16
+
25
x
2
4
d
x
\newline
Answer:
Get tutor help
Calculate the integral and write the answer in simplest form.
\newline
∫
(
3
x
2
+
6
x
4
)
d
x
\int\left(3 x^{2}+6 x^{4}\right) d x
∫
(
3
x
2
+
6
x
4
)
d
x
\newline
Answer:
Get tutor help
Calculate the integral and write the answer in simplest form.
\newline
∫
(
3
+
x
)
d
x
\int(3+x) d x
∫
(
3
+
x
)
d
x
\newline
Answer:
Get tutor help
Calculate the integral and write the answer in simplest form.
\newline
∫
(
−
2
x
+
5
x
5
)
d
x
\int\left(-2 x+5 x^{5}\right) d x
∫
(
−
2
x
+
5
x
5
)
d
x
\newline
Answer:
Get tutor help
Calculate the integral and write the answer in simplest form.
\newline
∫
(
−
6
x
−
3
+
2
x
)
d
x
\int\left(-6 x^{-3}+2 x\right) d x
∫
(
−
6
x
−
3
+
2
x
)
d
x
\newline
Answer:
Get tutor help
The function
s
(
t
)
s(t)
s
(
t
)
gives the number of students enrolled in a school by time
t
t
t
(in years).
\newline
What does
∫
15
18
s
′
(
t
)
d
t
=
20
\int_{15}^{18} s^{\prime}(t) d t=20
∫
15
18
s
′
(
t
)
d
t
=
20
mean?
\newline
Choose
1
1
1
answer:
\newline
(A) The rate of change of enrollment is
20
20
20
students per year more in year
18
18
18
than it was in year
15
15
15
.
\newline
(B) There were
20
20
20
more students enrolled in year
18
18
18
than in year
15
15
15
.
\newline
(C) Between years
15
15
15
and
18
18
18
, the cumulative number of years of schooling of all of the enrolled students is
20
20
20
years.
\newline
(D) There are
20
20
20
students enrolled in year
18
18
18
.
Get tutor help
The number of subscribers to a magazine is changing at a rate of
r
(
t
)
r(t)
r
(
t
)
subscribers per month (where
t
t
t
is time in months).
\newline
What does
∫
8
10
r
′
(
t
)
d
t
=
7
\int_{8}^{10} r^{\prime}(t) d t=7
∫
8
10
r
′
(
t
)
d
t
=
7
mean?
\newline
Choose
1
1
1
answer:
\newline
(A) The rate of change of number of subscribers increased by
7
7
7
subscribers per month between
t
=
8
t=8
t
=
8
and
t
=
10
t=10
t
=
10
months.
\newline
(B) As of month
10
10
10
, the magazine had
7
7
7
subscribers.
\newline
(C) The number of subscribers increased by
7
7
7
between
t
=
8
t=8
t
=
8
and
t
=
10
t=10
t
=
10
months.
\newline
(D) The average rate of change in subscribers between month
8
8
8
and month
10
10
10
was
7
7
7
subscribers per month.
Get tutor help
Consider the following problem:
\newline
The water level under a bridge is changing at a rate of
r
(
t
)
=
40
sin
(
π
t
6
)
r(t)=40 \sin \left(\frac{\pi t}{6}\right)
r
(
t
)
=
40
sin
(
6
π
t
)
centimeters per hour (where
t
t
t
is the time in hours). At time
t
=
3
t=3
t
=
3
, the water level is
91
91
91
centimeters. By how much does the water level change during the
4
th
4^{\text {th }}
4
th
hour?
\newline
Which expression can we use to solve the problem?
\newline
Choose
1
1
1
answer:
\newline
(A)
∫
0
4
r
(
t
)
d
t
\int_{0}^{4} r(t) d t
∫
0
4
r
(
t
)
d
t
\newline
(B)
∫
4
5
r
(
t
)
d
t
\int_{4}^{5} r(t) d t
∫
4
5
r
(
t
)
d
t
\newline
(C)
∫
3
4
r
(
t
)
d
t
\int_{3}^{4} r(t) d t
∫
3
4
r
(
t
)
d
t
\newline
(D)
∫
4
4
r
(
t
)
d
t
\int_{4}^{4} r(t) d t
∫
4
4
r
(
t
)
d
t
Get tutor help
A water tank is filled at a rate of
r
(
t
)
r(t)
r
(
t
)
liters per minute (where
t
t
t
is the time in minutes).
\newline
What does
∫
1
7
r
′
(
t
)
d
t
\int_{1}^{7} r^{\prime}(t) d t
∫
1
7
r
′
(
t
)
d
t
represent?
\newline
Choose
1
1
1
answer:
\newline
(A) The rate at which the tank was filled at
t
=
7
t=7
t
=
7
.
\newline
B) The average rate of filling between
t
=
1
t=1
t
=
1
and
t
=
7
t=7
t
=
7
.
\newline
(C) The change in the rate of filling between
t
=
1
t=1
t
=
1
and
t
=
7
t=7
t
=
7
.
\newline
(D) The amount of water filled between
t
=
1
t=1
t
=
1
and
t
=
7
t=7
t
=
7
.
Get tutor help
Consider the following problem:
\newline
The water level under a bridge is changing at a rate of
r
(
t
)
=
40
sin
(
π
t
6
)
r(t)=40 \sin \left(\frac{\pi t}{6}\right)
r
(
t
)
=
40
sin
(
6
π
t
)
centimeters per hour (where
t
t
t
is the time in hours). At time
t
=
3
t=3
t
=
3
, the water level is
91
91
91
centimeters. By how much does the water level change during the
4
th
4^{\text {th }}
4
th
hour?
\newline
Which expression can we use to solve the problem?
\newline
Choose
1
1
1
answer:
\newline
(A)
∫
3
4
r
(
t
)
d
t
\int_{3}^{4} r(t) d t
∫
3
4
r
(
t
)
d
t
\newline
(B)
∫
0
4
r
(
t
)
d
t
\int_{0}^{4} r(t) d t
∫
0
4
r
(
t
)
d
t
\newline
(C)
∫
4
5
r
(
t
)
d
t
\int_{4}^{5} r(t) d t
∫
4
5
r
(
t
)
d
t
\newline
(D)
∫
4
4
r
(
t
)
d
t
\int_{4}^{4} r(t) d t
∫
4
4
r
(
t
)
d
t
Get tutor help
Previous
1
2
