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Math Problems
Calculus
Find indefinite integrals using the substitution
Use the laws to re-write the following expression as a power of
x
x
x
:
\newline
x
x
x
x
\sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}}
x
x
x
x
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What is the constant term in the expression
\newline
(
1
)
/
(
12
)
(
x
−
4
)
−
(
5
)
/
(
2
)
x
(
x
2
−
7
x
+
1
)
(1)/(12)(x-4)-(5)/(2)x(x^{2}-7x+1)
(
1
)
/
(
12
)
(
x
−
4
)
−
(
5
)
/
(
2
)
x
(
x
2
−
7
x
+
1
)
?
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Simplify the expression.
\newline
y
2
−
25
y
2
−
10
y
+
25
\frac{y^{2}-25}{y^{2}-10 y+25}
y
2
−
10
y
+
25
y
2
−
25
\newline
Enclose numerators and denominators in parentheses. For example,
(
a
−
b
)
/
(
1
+
n
)
(a-b) /(1+n)
(
a
−
b
)
/
(
1
+
n
)
.
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∫
5
2
x
+
1
d
x
\int \frac{5}{2 x+1} d x
∫
2
x
+
1
5
d
x
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The function
g
(
x
)
g(x)
g
(
x
)
is odd and continuous for all
x
\mathrm{x}
x
. If
∫
0
a
g
(
x
)
d
x
=
3.5
\int_{0}^{a} g(x) d x=3.5
∫
0
a
g
(
x
)
d
x
=
3.5
, what is
∫
−
a
a
g
(
x
)
d
x
?
\int_{-a}^{a} g(x) d x ?
∫
−
a
a
g
(
x
)
d
x
?
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An irrational number between
2
\sqrt{2}
2
and
3
\sqrt{3}
3
is
\newline
(a)
(
2
+
3
)
(\sqrt{2}+\sqrt{3})
(
2
+
3
)
\newline
(b)
2
×
3
\sqrt{2} \times \sqrt{3}
2
×
3
\newline
(c)
5
1
/
4
5^{1 / 4}
5
1/4
\newline
(d)
6
1
/
4
6^{1 / 4}
6
1/4
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Write the equation of all horizontal asymptotes of the function
f
(
x
)
=
6
x
−
e
x
3
x
−
2
x
2
f(x)=\frac{6x-e^{x}}{3x-2x^{2}}
f
(
x
)
=
3
x
−
2
x
2
6
x
−
e
x
.
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Find the truth set of the following simultaneous equation
\newline
5
x
6
−
3
4
\frac{5x}{6}-\frac{3}{4}
6
5
x
−
4
3
y
=
1
2
×
2
3
y=\frac{1}{2}\times\frac{2}{3}
y
=
2
1
×
3
2
y
=
5
2
y=\frac{5}{2}
y
=
2
5
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Use Green's theorem to evaluate the line integral along the given positively oriented curve,
∫
C
7
y
3
d
x
−
7
x
3
d
y
,
is the circle
x
2
+
y
2
=
4
\int_{C}7y^{3}\,dx-7x^{3}\,dy,\quad \text{is the circle } x^{2}+y^{2}=4
∫
C
7
y
3
d
x
−
7
x
3
d
y
,
is the circle
x
2
+
y
2
=
4
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What is the simplified form of the polynomial expression shown?
\newline
−
3
x
2
(
x
−
y
2
)
−
(
y
3
−
5
)
−
3
y
2
(
x
2
−
4
y
)
+
4
x
3
-3x^{2}(x-y^{2})-(y^{3}-5)-3y^{2}(x^{2}-4y)+4x^{3}
−
3
x
2
(
x
−
y
2
)
−
(
y
3
−
5
)
−
3
y
2
(
x
2
−
4
y
)
+
4
x
3
\newline
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Which of the following is a rational number?
\newline
Choices:
\newline
(A)
π
\pi
π
\newline
(B)
4
7
\frac{4}{7}
7
4
\newline
(C)
10
\sqrt{10}
10
\newline
(D)
3
\sqrt{3}
3
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∫
4
x
cos
(
2
−
3
x
)
d
x
\int 4 x \cos (2-3 x) d x
∫
4
x
cos
(
2
−
3
x
)
d
x
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(b) Use integration by parts to evaluate
∫
0
π
2
x
⋅
cos
x
d
x
\int_0^{\frac{\pi}{2}} x \cdot \cos x \, dx
∫
0
2
π
x
⋅
cos
x
d
x
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∫
sin
2
x
sin
x
d
x
\int \frac{\sin 2 x}{\sin x} d x
∫
s
i
n
x
s
i
n
2
x
d
x
=
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∫
x
3
−
2
x
x
d
x
\int \frac{x^{3}-2 \sqrt{x}}{x} d x
∫
x
x
3
−
2
x
d
x
=
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Simplify:
∫
x
x
d
x
\int \frac{x}{x}dx
∫
x
x
d
x
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∫
sec
t
(
sec
t
+
tan
t
)
d
t
\int \sec t(\sec t+\tan t) d t
∫
sec
t
(
sec
t
+
tan
t
)
d
t
=
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What could be the value of
x
x
x
in the following equation? Select all that apply.
\newline
x
2
=
1
36
x^2 = \frac{1}{36}
x
2
=
36
1
\newline
Multi-select Choices:
\newline
(A)
1
18
\frac{1}{18}
18
1
\newline
(B)
1
36
\sqrt{\frac{1}{36}}
36
1
\newline
(C)
−
1
18
-\frac{1}{18}
−
18
1
\newline
(D)
−
1
36
-\sqrt{\frac{1}{36}}
−
36
1
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RT
4
4
4
- Mathematical Knowledge
\newline
Evaluate
\newline
(
b
−
a
)
2
−
3
c
(
−
a
)
3
\frac{(b-a)^{2}-3c}{(-a)^{3}}
(
−
a
)
3
(
b
−
a
)
2
−
3
c
, when
\newline
a
=
4
a=4
a
=
4
,
b
=
5
b=5
b
=
5
, and
\newline
c
=
2
c=2
c
=
2
\newline
A
5
12
\frac{5}{12}
12
5
\newline
B
−
5
12
-\frac{5}{12}
−
12
5
\newline
C
5
64
\frac{5}{64}
64
5
\newline
D
−
5
64
\quad-\frac{5}{64}
−
64
5
\newline
Click the button or type the letter to the left of you
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Find the
8
th
8^{\text{th}}
8
th
term in the sequence
\newline
−
1
2
,
−
1
,
−
2
,
−
4
,
…
-\frac{1}{2},-1,-2,-4,\dots
−
2
1
,
−
1
,
−
2
,
−
4
,
…
\newline
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∫
1
x
3
ln
x
d
x
\int \frac{1}{x^{3}}\ln x \, dx
∫
x
3
1
ln
x
d
x
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Evaluate the integral
∫
Γ
(
e
2
2
⋅
(
2
2
+
1
)
)
d
z
\int_{\Gamma}\left(\frac{e^{2}}{2\cdot(2^{2}+1)}\right)dz
∫
Γ
(
2
⋅
(
2
2
+
1
)
e
2
)
d
z
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Evaluate the integral
∫
x
+
3
4
x
+
4
d
x
\int \frac{x+3}{4 x+4} d x
∫
4
x
+
4
x
+
3
d
x
.
\newline
Choose
1
1
1
answer:
\newline
(A)
1
4
x
+
ln
∣
x
+
1
∣
+
C
\frac{1}{4} x+\ln |x+1|+C
4
1
x
+
ln
∣
x
+
1∣
+
C
\newline
(B)
1
4
x
+
2
ln
∣
x
+
1
∣
+
C
\frac{1}{4} x+2 \ln |x+1|+C
4
1
x
+
2
ln
∣
x
+
1∣
+
C
\newline
(C)
1
4
x
+
ln
∣
x
+
1
∣
2
+
C
\frac{1}{4} x+\frac{\ln |x+1|}{2}+C
4
1
x
+
2
l
n
∣
x
+
1∣
+
C
\newline
(D)
1
4
x
+
ln
∣
x
+
1
∣
4
+
C
\frac{1}{4} x+\frac{\ln |x+1|}{4}+C
4
1
x
+
4
l
n
∣
x
+
1∣
+
C
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g
(
x
)
=
∫
2
x
13
1
+
t
2
d
t
g
′
(
5
)
=
\begin{array}{l}g(x)=\int_{2}^{x} \frac{13}{1+t^{2}} d t \\ g^{\prime}(5)=\end{array}
g
(
x
)
=
∫
2
x
1
+
t
2
13
d
t
g
′
(
5
)
=
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If
I
=
∫
d
x
sin
(
x
−
a
)
sin
(
x
−
b
)
I=\int \frac{\mathrm{d}x}{\sin(x-a)\sin(x-b)}
I
=
∫
s
i
n
(
x
−
a
)
s
i
n
(
x
−
b
)
d
x
, then
\newline
I
I
I
is given by
Get tutor help
\newline
The value of
\newline
∫
e
x
(
x
2
+
4
x
+
4
(
x
+
4
)
2
)
d
x
\int e^{x}\left(\frac{x^{2}+4x+4}{(x+4)^{2}}\right)dx
∫
e
x
(
(
x
+
4
)
2
x
2
+
4
x
+
4
)
d
x
is
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what is the range of
cos
x
\cos{\sqrt{x}}
cos
x
where
.
{.}
.
represents fractional part
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The function
\newline
f
(
x
)
=
4
−
x
2
4
x
−
x
3
f(x)=\frac{4-x^{2}}{4x-x^{3}}
f
(
x
)
=
4
x
−
x
3
4
−
x
2
is
\newline
(a) discontinuous at only one point
\newline
(b) discontinuous at exactly two points
\newline
(c) discontinuous at exactly three points
\newline
(d) None of the above
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Write the following as an exponential expression.
\newline
1
3
3
4
\sqrt[4]{13^{3}}
4
1
3
3
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∫
x
(
x
−
1
)
5
d
x
\int x(x-1)^{5} d x
∫
x
(
x
−
1
)
5
d
x
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∫
x
−
3
(
x
−
1
)
(
x
−
2
)
d
x
\int \frac{x-3}{(x-1)(x-2)} d x
∫
(
x
−
1
)
(
x
−
2
)
x
−
3
d
x
Get tutor help
Solve the system of equations.
\newline
y
=
−
4
x
y
=
2
x
2
−
15
x
\begin{array}{l} y=-4 x \\ y=2 x^{2}-15 x \end{array}
y
=
−
4
x
y
=
2
x
2
−
15
x
\newline
solutions
=
(
= (
=
(
□
\square
□
and
□
)
\square )
□
)
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Distribute to create an equivalent expression with the fewest symbols possible.
\newline
1
2
(
10
x
+
20
y
+
10
z
)
=
□
\frac{1}{2}(10 x+20 y+10 z) = \square
2
1
(
10
x
+
20
y
+
10
z
)
=
□
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Given that
y
=
27
x
3
+
1
x
+
1
y=27 x^{3}+\frac{1}{x}+1
y
=
27
x
3
+
x
1
+
1
\newline
(a) write down an expression for
d
y
d
x
\frac{\mathrm{d} y}{\mathrm{~d} x}
d
x
d
y
,
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∫
7
(
x
−
2
)
(
x
+
5
)
d
x
\int \frac{7}{(x-2)(x+5)} d x
∫
(
x
−
2
)
(
x
+
5
)
7
d
x
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d
=
(
−
11
−
(
−
2
)
2
+
(
10
−
7
)
2
d
=
\begin{array}{c}d=\sqrt{\left(-11-(-2)^{2}+(10-7)^{2}\right.} \\ d=\end{array}
d
=
(
−
11
−
(
−
2
)
2
+
(
10
−
7
)
2
d
=
Get tutor help
Calculate the iterated integral.
\newline
∫
0
1
∫
0
1
5
s
+
t
d
s
d
t
\int_{0}^{1} \int_{0}^{1} 5 \sqrt{s+t} d s d t
∫
0
1
∫
0
1
5
s
+
t
d
s
d
t
Get tutor help
Simplify the following expression.
\newline
(
4
y
+
7
)
(
y
−
2
)
4
y
2
+
□
y
+
□
\begin{array}{l} (4 y+7)(y-2) \\ 4 y^{2}+\square y+\square \end{array}
(
4
y
+
7
)
(
y
−
2
)
4
y
2
+
□
y
+
□
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Given the following point on the unit circle, find the angle, to the nearest tenth of a degree (if necessary), of the terminal side through that point,
0
∘
≤
θ
<
36
0
∘
0^{\circ} \leq \theta<360^{\circ}
0
∘
≤
θ
<
36
0
∘
.
\newline
P
=
(
−
10
4
,
−
6
4
)
P=\left(-\frac{\sqrt{10}}{4},-\frac{\sqrt{6}}{4}\right)
P
=
(
−
4
10
,
−
4
6
)
\newline
Answer:
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∫
14
x
3
/
2
−
49
x
d
x
\int \frac{14}{x^{3 / 2}-49 \sqrt{x}} d x
∫
x
3/2
−
49
x
14
d
x
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∫
x
x
+
1
\int \frac{\sqrt{x}}{x+1}
∫
x
+
1
x
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∫
2
x
(
x
−
1
)
(
x
−
2
)
(
x
+
4
)
d
x
\int \frac{2 x}{(x-1)(x-2)(x+4)} d x
∫
(
x
−
1
)
(
x
−
2
)
(
x
+
4
)
2
x
d
x
=
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∫
9
−
x
2
x
4
d
x
\int \frac{\sqrt{9-x^{2}}}{x^{4}} d x
∫
x
4
9
−
x
2
d
x
=
Get tutor help
Using the substitution
x
=
sin
2
θ
x=\sin^{2}\theta
x
=
sin
2
θ
, or otherwise, evaluate
∫
0
1
2
x
1
−
x
d
x
\int_{0}^{\frac{1}{2}}\sqrt{\frac{x}{1-x}}dx
∫
0
2
1
1
−
x
x
d
x
.
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(e) Use the substitution
\newline
x
=
3
sin
θ
x=3\sin \theta
x
=
3
sin
θ
to evaluate
\newline
∫
0
3
2
d
x
(
9
−
x
2
)
3
2
\int_{0}^{\frac{3}{\sqrt{2}}}\frac{dx}{(9-x^{2})^{\frac{3}{2}}}
∫
0
2
3
(
9
−
x
2
)
2
3
d
x
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What is the general solution to the differential equation that generated the slope field?
\newline
Choose
1
1
1
answer:
\newline
(
a
)
(a)
(
a
)
y
=
x
+
C
y=x+C
y
=
x
+
C
\newline
(
b
)
(b)
(
b
)
y
=
x
2
+
C
y=x^{2}+C
y
=
x
2
+
C
\newline
(
c
)
(c)
(
c
)
x
=
y
2
+
C
x=y^{2}+C
x
=
y
2
+
C
\newline
(
d
)
(d)
(
d
)
y
2
=
x
2
+
C
y^{2}=x^{2}+C
y
2
=
x
2
+
C
\newline
(
e
)
(e)
(
e
)
x
2
+
y
2
=
C
x^{2}+y^{2}=C
x
2
+
y
2
=
C
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Which of the following is a rational number?
\newline
Choices:
\newline
(A)
2
7
\frac{2}{7}
7
2
\newline
(B)
π
\pi
π
\newline
(C)
3
\sqrt{3}
3
\newline
(D)
8
\sqrt{8}
8
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Integrate
∫
d
x
7
x
2
+
8
\int \frac{d x}{7 x^{2}+8}
∫
7
x
2
+
8
d
x
=
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Rewrite the function by using long division to perform
(
110
x
−
11
,
000
)
÷
(
x
−
150
)
(110 x-11,000) \div(x-150)
(
110
x
−
11
,
000
)
÷
(
x
−
150
)
.
\newline
(
110
x
−
11
,
000
)
÷
(
x
−
150
)
=
□
(110 x-11,000) \div(x-150)=\square
(
110
x
−
11
,
000
)
÷
(
x
−
150
)
=
□
\newline
(Simplify your answer. If there is a remainder, type your answer in the form quotient
+
remainder
divisor
+\frac{\text { remainder }}{\text { divisor }}
+
divisor
remainder
.)
\newline
Then use this new form of the function to find
f
(
40
)
\mathrm{f}(40)
f
(
40
)
.
\newline
f
(
40
)
=
□
f(40)=\square
f
(
40
)
=
□
(Round to the nearest integer as needed.)
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Evaluate the integral:
∫
x
5
−
x
4
−
3
x
+
5
(
x
2
−
x
+
1
)
2
\int \frac{x^{5}-x^{4}-3 x+5}{\left(x^{2}-x+1\right)^{2}}
∫
(
x
2
−
x
+
1
)
2
x
5
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