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Math Problems
Calculus
Evaluate definite integrals using the chain rule
∫
0
1
x
d
x
x
4
+
1
=
\int_{0}^{1}\frac{x\,dx}{\sqrt{x^{4}+1}}=
∫
0
1
x
4
+
1
x
d
x
=
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9
9
9
.
∫
0
1
d
x
4
x
2
+
4
x
+
5
\int_{0}^{1} \frac{d x}{4 x^{2}+4 x+5}
∫
0
1
4
x
2
+
4
x
+
5
d
x
.
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∫
x
2
+
2
x
x
−
3
d
x
\int \frac{x^{2}+2 x}{x-3} d x
∫
x
−
3
x
2
+
2
x
d
x
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∫
0
3
∫
0
−
x
+
5
y
2
−
x
d
y
d
x
\int_{0}^{3} \int_{0}^{-x+5} y^{2}-x d y d x
∫
0
3
∫
0
−
x
+
5
y
2
−
x
d
y
d
x
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∫
1
e
−
x
−
1
d
x
\int \frac{1}{e^{-x}-1} d x
∫
e
−
x
−
1
1
d
x
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∫
x
2
d
x
x
2
−
4
\int \frac{x^{2} d x}{x^{2}-4}
∫
x
2
−
4
x
2
d
x
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17
17
17
.
lim
(
x
,
y
)
→
(
0
,
0
)
x
2
+
y
2
x
2
+
y
2
+
1
−
1
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1}
lim
(
x
,
y
)
→
(
0
,
0
)
x
2
+
y
2
+
1
−
1
x
2
+
y
2
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If
f
(
x
)
=
(
x
2
+
1
)
3
f(x)=\left(x^{2}+1\right)^{3}
f
(
x
)
=
(
x
2
+
1
)
3
, what is
lim
x
→
−
1
f
(
x
)
−
f
(
−
1
)
x
+
1
?
\lim _{x \rightarrow-1} \frac{f(x)-f(-1)}{x+1} ?
lim
x
→
−
1
x
+
1
f
(
x
)
−
f
(
−
1
)
?
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2
2
2
. Find:
\newline
(a)
∫
13
e
x
d
x
\int 13 e^{x} d x
∫
13
e
x
d
x
\newline
(d)
∫
3
e
−
(
2
x
+
7
)
d
x
\int 3 e^{-(2 x+7)} d x
∫
3
e
−
(
2
x
+
7
)
d
x
\newline
(b)
∫
(
3
e
x
+
4
x
)
d
x
(
x
>
0
)
\int\left(3 e^{x}+\frac{4}{x}\right) d x \quad(x>0)
∫
(
3
e
x
+
x
4
)
d
x
(
x
>
0
)
\newline
(e)
∫
4
x
e
x
2
+
3
d
x
\int 4 x e^{x^{2}+3} d x
∫
4
x
e
x
2
+
3
d
x
\newline
(c)
∫
(
5
e
x
+
3
x
2
)
d
x
(
x
≠
0
)
\int\left(5 e^{x}+\frac{3}{x^{2}}\right) d x \quad(x \neq 0)
∫
(
5
e
x
+
x
2
3
)
d
x
(
x
=
0
)
\newline
(f)
∫
x
e
x
2
!
9
d
x
\int x e^{x^{2}!9} d x
∫
x
e
x
2
!
9
d
x
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Edit
\newline
3
3
3
\newline
Assignment
4
4
4
\newline
1
1
1
. Find the following:
\newline
(a)
∫
16
x
−
3
d
x
(
x
≠
0
)
\int 16 x^{-3} d x \quad(x \neq 0)
∫
16
x
−
3
d
x
(
x
=
0
)
\newline
(d)
∫
2
e
−
2
x
d
x
\int 2 e^{-2 x} d x
∫
2
e
−
2
x
d
x
\newline
(b)
∫
9
x
8
d
x
\int 9 x^{8} d x
∫
9
x
8
d
x
\newline
(e)
∫
4
x
x
2
+
1
d
x
\int \frac{4 x}{x^{2}+1} d x
∫
x
2
+
1
4
x
d
x
\newline
(c)
∫
(
x
5
−
3
x
)
d
x
\int\left(x^{5}-3 x\right) d x
∫
(
x
5
−
3
x
)
d
x
\newline
(f)
∫
(
2
a
x
+
b
)
(
a
x
2
+
b
x
)
7
d
x
\int(2 a x+b)\left(a x^{2}+b x\right)^{7} d x
∫
(
2
a
x
+
b
)
(
a
x
2
+
b
x
)
7
d
x
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∫
1
4
u
−
2
u
d
u
\int_{1}^{4} \frac{u-2}{\sqrt{u}} d u
∫
1
4
u
u
−
2
d
u
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24
24
24
.
∫
1
4
(
3
−
∣
x
−
3
∣
)
d
x
\int_{1}^{4}(3-|x-3|) d x
∫
1
4
(
3
−
∣
x
−
3∣
)
d
x
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∫
−
2
1
(
2
−
∣
x
∣
)
d
x
\int_{-2}^{1}(2-|x|) d x
∫
−
2
1
(
2
−
∣
x
∣
)
d
x
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∫
π
4
π
cos
(
2
θ
)
d
θ
\int_{\frac{\pi}{4}}^{\pi} \cos (2 \theta) d \theta
∫
4
π
π
cos
(
2
θ
)
d
θ
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π
4
∫
π
cos
(
2
θ
)
d
θ
\frac{\pi}{4} \int^{\pi} \cos (2 \theta) d \theta
4
π
∫
π
cos
(
2
θ
)
d
θ
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∫
−
π
6
π
3
−
sin
x
cos
2
x
d
x
\int_{-\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{-\sin x}{\cos ^{2} x} d x
∫
−
6
π
3
π
c
o
s
2
x
−
s
i
n
x
d
x
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∫
−
π
6
π
3
−
sin
x
cos
2
x
d
x
\int_{-\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{-\sin x}{\cos ^{2} x} d x
∫
−
6
π
3
π
cos
2
x
−
sin
x
d
x
\newline
Click to view the table of general integ
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17
17
17
.
\newline
Write the following as a sum of logarithms:
\newline
log
(
x
9
y
2
z
2
)
=
\log \left(x^{9} y^{2} z^{2}\right)=
lo
g
(
x
9
y
2
z
2
)
=
\newline
□
\square
□
log
(
x
)
+
\log (x)+
lo
g
(
x
)
+
□
\square
□
log
(
y
)
+
\log (y)+
lo
g
(
y
)
+
□
\square
□
log
(
z
)
\log (z)
lo
g
(
z
)
\newline
Nextitem
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r
(
x
)
=
∫
−
2
x
2
t
e
4
t
2
d
h
r(x)=\int_{-2}^{x}2te^{4t^{2}}\,dh
r
(
x
)
=
∫
−
2
x
2
t
e
4
t
2
d
h
Get tutor help
∫
0
π
2
x
cos
x
d
x
\int_{0}^{\pi} 2 x \cos x d x
∫
0
π
2
x
cos
x
d
x
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∫
d
x
(
9
+
x
2
)
3
/
2
\int \frac{d x}{\left(9+x^{2}\right)^{3 / 2}}
∫
(
9
+
x
2
)
3/2
d
x
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∫
x
2
d
x
(
x
2
+
16
)
2
\int \frac{x^{2} d x}{\left(x^{2}+16\right)^{2}}
∫
(
x
2
+
16
)
2
x
2
d
x
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∫
−
1
2
x
e
6
x
d
x
\int_{-1}^{2} x e^{6 x} d x
∫
−
1
2
x
e
6
x
d
x
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∫
x
5
x
3
+
1
d
x
\int x^{5} \sqrt{x^{3}+1} d x
∫
x
5
x
3
+
1
d
x
Get tutor help
∫
(
3
x
+
4
)
cos
x
d
x
∫
x
5
x
3
+
1
d
x
\begin{array}{l}\int(3 x+4) \cos x d x \\ \int x^{5} \sqrt{x^{3}+1} d x\end{array}
∫
(
3
x
+
4
)
cos
x
d
x
∫
x
5
x
3
+
1
d
x
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Find the following definite integral.
\newline
∫
−
1
0
2
x
d
x
\int_{-1}^{0} 2 x d x
∫
−
1
0
2
x
d
x
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∫
x
4
+
1
x
2
−
1
d
x
\int \frac{x^{4}+1}{x^{2}-1} d x
∫
x
2
−
1
x
4
+
1
d
x
Get tutor help
∫
x
2
−
1
x
2
x
2
−
1
d
x
\int \frac{\sqrt{x^{2}-1}}{x^{2} \sqrt{x^{2}-1}} d x
∫
x
2
x
2
−
1
x
2
−
1
d
x
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∫
x
2
+
1
x
2
x
2
−
1
d
x
\int \frac{\sqrt{x^{2}+1}}{x^{2} \sqrt{x^{2}-1}} d x
∫
x
2
x
2
−
1
x
2
+
1
d
x
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∫
t
e
t
1
+
e
2
t
d
t
\int \frac{t e^{t}}{1+e^{2 t}} d t
∫
1
+
e
2
t
t
e
t
d
t
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∫
1
x
3
+
1
d
x
\int \frac{1}{x^{3}+1} \, dx
∫
x
3
+
1
1
d
x
Get tutor help
Select all the expressions that are equivalent to
6
−
4
×
6
−
4
6^{-4} \times 6^{-4}
6
−
4
×
6
−
4
.
\newline
Multi-select Choices:
\newline
(A)
1
6
16
\frac{1}{6^{16}}
6
16
1
\newline
(B)
6
16
6^{16}
6
16
\newline
(C)
1
6
−
8
\frac{1}{6^{-8}}
6
−
8
1
\newline
(D)
6
−
8
6^{-8}
6
−
8
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∫
x
1
3
cos
(
x
2
/
3
,
8
)
d
x
\int x^{\frac{1}{3}} \cos \left(x^{2 / 3}, 8\right) d x
∫
x
3
1
cos
(
x
2/3
,
8
)
d
x
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∫
0
π
x
sin
x
1
+
cos
2
x
d
x
\int_{0}^{\pi} \frac{x \sin x}{1+\cos ^{2} x} d x
∫
0
π
1
+
c
o
s
2
x
x
s
i
n
x
d
x
Get tutor help
1
1
1
. Noma'lum burchakni toping (
1
1
1
-rasm).
\newline
2
2
2
. Uchburchakning tashqi burchagi
12
0
∘
120^{\circ}
12
0
∘
bo'lib, unga qo'shni bo'Imagan ichki burchaklari
1
1
1
:
2
2
2
nisbatda bo'lsa, uchburchakning burchaklarini toping.
\newline
3
3
3
. Agar
2
2
2
-rasmda
∠
A
C
B
=
9
0
∘
,
C
D
=
B
D
\angle A C B=90^{\circ}, C D=B D
∠
A
CB
=
9
0
∘
,
C
D
=
B
D
va
A
B
=
24
s
m
A B=24 \mathrm{sm}
A
B
=
24
sm
bo
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∫
0
2
2
x
(
x
2
2
+
3
)
5
d
x
\int_{0}^{2} 2 x\left(\frac{x^{2}}{2}+3\right)^{5} d x
∫
0
2
2
x
(
2
x
2
+
3
)
5
d
x
Get tutor help
savasrealize.com/dashboard/classes/
64
64
64
c
0
0
0
d
3292
3292
3292
e
855
855
855
b
71
71
71
a
5
5
5
d
56
56
56
e
7
7
7
e/details/assignments/
0726
0726
0726
fcb
73
73
73
c
794
794
794
ad
696
696
696
c
\newline
Maps
\newline
SCHLOETER HOMEROOM
502
502
502
8
8
8
(A)
\newline
Topic
8
8
8
: Online Assessment Copy
1
1
1
\newline
A. Select all the expressions that show breaking the addends
7
5
12
7 \frac{5}{12}
7
12
5
and
11
2
3
11 \frac{2}{3}
11
3
2
apart to make a whole.
\newline
7
+
5
12
+
11
+
2
3
7+\frac{5}{12}+11+\frac{2}{3}
7
+
12
5
+
11
+
3
2
\newline
7
+
5
12
+
8
12
+
11
7+\frac{5}{12}+\frac{8}{12}+11
7
+
12
5
+
12
8
+
11
\newline
7
+
5
12
+
7
12
+
11
1
12
7+\frac{5}{12}+\frac{7}{12}+11 \frac{1}{12}
7
+
12
5
+
12
7
+
11
12
1
\newline
7
+
5
12
+
5
12
+
11
3
12
7+\frac{5}{12}+\frac{5}{12}+11 \frac{3}{12}
7
+
12
5
+
12
5
+
11
12
3
\newline
7
+
1
12
+
4
12
+
8
12
+
11
7+\frac{1}{12}+\frac{4}{12}+\frac{8}{12}+11
7
+
12
1
+
12
4
+
12
8
+
11
Get tutor help
Let
H
(
x
)
=
3
x
+
∫
1
x
2
g
(
x
)
d
x
H(x)=3 x+\int_{1}^{x^{2}} g(x) d x
H
(
x
)
=
3
x
+
∫
1
x
2
g
(
x
)
d
x
\newline
(c) Find
H
′
(
2
)
H^{\prime}(2)
H
′
(
2
)
and
H
′
′
(
2
)
H^{\prime \prime}(2)
H
′′
(
2
)
.
Get tutor help
∫
0
π
2
sin
2
θ
(
1
+
cos
θ
)
2
d
θ
\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{2} \theta}{(1+\cos \theta)^{2}} d \theta
∫
0
2
π
(
1
+
c
o
s
θ
)
2
s
i
n
2
θ
d
θ
Get tutor help
iolve the Definite Integral:
\newline
∫
−
6
14
10
x
4
+
12
x
3
−
16
x
2
−
9
x
+
15
d
x
\int_{-6}^{14} 10 x^{4}+12 x^{3}-16 x^{2}-9 x+15 d x
∫
−
6
14
10
x
4
+
12
x
3
−
16
x
2
−
9
x
+
15
d
x
Get tutor help
Solve the Definite Integral:
\newline
∫
−
6
14
10
x
4
+
12
x
3
−
16
x
2
−
9
x
+
15
d
x
\int_{-6}^{14} 10 x^{4}+12 x^{3}-16 x^{2}-9 x+15 d x
∫
−
6
14
10
x
4
+
12
x
3
−
16
x
2
−
9
x
+
15
d
x
\newline
Section
5
5
5
.
4
4
4
Get tutor help
2
2
2
.
∫
0
1
2
π
2
cos
2
t
d
t
\int_{0}^{\frac{1}{2} \pi^{2}} \cos \sqrt{2 t} d t
∫
0
2
1
π
2
cos
2
t
d
t
Get tutor help
If
∫
3
14
f
(
x
)
d
x
=
68
\int_{3}^{14} f(x) d x=68
∫
3
14
f
(
x
)
d
x
=
68
and
∫
11
14
f
(
x
)
d
x
=
10
\int_{11}^{14} f(x) d x=10
∫
11
14
f
(
x
)
d
x
=
10
, calculate
∫
3
11
f
(
x
)
d
x
\int_{3}^{11} f(x) d x
∫
3
11
f
(
x
)
d
x
Get tutor help
∫
−
2
2
(
(
x
3
cos
x
2
+
1
2
)
4
−
x
2
d
x
\int_{-2}^{2}\left(\left(x^{3} \cos \frac{x}{2}+\frac{1}{2}\right) \sqrt{4-x^{2}} d x\right.
∫
−
2
2
(
(
x
3
cos
2
x
+
2
1
)
4
−
x
2
d
x
Get tutor help
find
∫
0
1
(
f
(
x
)
−
g
(
x
)
)
d
x
\int_{0}^{1}(f(x)-g(x))\,dx
∫
0
1
(
f
(
x
)
−
g
(
x
))
d
x
, if
∫
0
1
(
f
(
x
)
−
2
g
(
x
)
)
d
x
=
6
\int_{0}^{1}(f(x)-2g(x))\,dx=6
∫
0
1
(
f
(
x
)
−
2
g
(
x
))
d
x
=
6
,
∫
0
1
(
2
f
(
x
)
+
2
g
(
x
)
)
d
x
=
9
\int_{0}^{1}(2f(x)+2g(x))\,dx=9
∫
0
1
(
2
f
(
x
)
+
2
g
(
x
))
d
x
=
9
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Calculate the definite integral of the function
f
(
x
)
=
2
x
f(x) = 2x
f
(
x
)
=
2
x
from
x
=
1
x = 1
x
=
1
to
x
=
3
x = 3
x
=
3
.
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∫
0
1
cos
π
1
−
x
⋅
d
x
(
1
−
x
)
2
\int_{0}^{1} \cos \frac{\pi}{1-x} \cdot \frac{d x}{(1-x)^{2}}
∫
0
1
cos
1
−
x
π
⋅
(
1
−
x
)
2
d
x
.
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Вычислить несобственный интеграл или установить его расходимость
\newline
∫
0
1
cos
π
1
−
x
⋅
d
x
(
1
−
x
)
2
\int_{0}^{1} \cos \frac{\pi}{1-x} \cdot \frac{d x}{(1-x)^{2}}
∫
0
1
cos
1
−
x
π
⋅
(
1
−
x
)
2
d
x
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∫
0
4
x
1
+
2
x
d
x
\int_{0}^{4} \frac{x}{\sqrt{1+2 x}} d x
∫
0
4
1
+
2
x
x
d
x
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1
1
1
. Вычислить определённый интеграл
\newline
∫
0
3
4
d
x
(
x
+
1
)
x
2
+
1
\int_{0}^{\frac{3}{4}} \frac{d x}{(x+1) \sqrt{x^{2}+1}}
∫
0
4
3
(
x
+
1
)
x
2
+
1
d
x
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1
2
3
...
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