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Math Problems
Calculus
Evaluate definite integrals using the chain rule
Integrate.
\newline
∫
0
1
(
3
x
3
−
x
2
+
2
x
−
4
)
x
2
−
3
x
+
2
d
x
\int_{0}^{1} \frac{\left(3 x^{3}-x^{2}+2 x-4\right)}{\sqrt{x^{2}-3 x+2}} d x
∫
0
1
x
2
−
3
x
+
2
(
3
x
3
−
x
2
+
2
x
−
4
)
d
x
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Integrate.
\newline
∫
20000
(
x
+
100
)
3
d
x
\int \frac{20000}{(x+100)^{3}} d x
∫
(
x
+
100
)
3
20000
d
x
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i)
lim
x
→
∞
x
2
x
3
−
5
\lim _{x \rightarrow \infty} \frac{x^{2}}{x^{3}-5}
lim
x
→
∞
x
3
−
5
x
2
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∫
−
2
2
(
x
3
cos
x
2
+
1
2
)
4
−
x
2
d
x
\int_{-2}^{2}\left(x^{3} \cos \frac{x}{2}+\frac{1}{2}\right) \sqrt{4-x^{2} d x}
∫
−
2
2
(
x
3
cos
2
x
+
2
1
)
4
−
x
2
d
x
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Integrate.
\newline
∫
−
2
2
(
x
3
cos
(
x
2
)
+
1
2
)
4
−
x
2
d
x
\int_{-2}^{2}\left(x^{3} \cos \left(\frac{x}{2}\right)+\frac{1}{2}\right) \sqrt{4-x^{2}} d x
∫
−
2
2
(
x
3
cos
(
2
x
)
+
2
1
)
4
−
x
2
d
x
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Find
∫
0
2
f
(
x
,
y
)
d
x
\int_{0}^{2} f(x, y) d x
∫
0
2
f
(
x
,
y
)
d
x
and
∫
0
3
f
(
x
,
y
)
d
y
\int_{0}^{3} f(x, y) d y
∫
0
3
f
(
x
,
y
)
d
y
.
\newline
f
(
x
,
y
)
=
11
y
x
+
2
∫
0
2
f
(
x
,
y
)
d
x
=
□
∫
0
3
f
(
x
,
y
)
d
y
=
□
\begin{array}{r} f(x, y)=11 y \sqrt{x+2} \\ \int_{0}^{2} f(x, y) d x=\square \\ \int_{0}^{3} f(x, y) d y=\square \end{array}
f
(
x
,
y
)
=
11
y
x
+
2
∫
0
2
f
(
x
,
y
)
d
x
=
□
∫
0
3
f
(
x
,
y
)
d
y
=
□
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What is the product of
(
1
−
p
)
(1 - p)
(
1
−
p
)
and
(
1
2
−
p
)
(\frac{1}{2} - p)
(
2
1
−
p
)
all reduced by
p
p
p
?
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Compute
\newline
F
(
α
)
=
∫
0
π
/
2
ln
(
α
2
−
sin
(
t
)
2
)
d
t
F(\alpha)=\int_{0}^{\pi/2}\ln(\alpha^{2}-\sin(t)^{2})dt
F
(
α
)
=
∫
0
π
/2
ln
(
α
2
−
sin
(
t
)
2
)
d
t
\newline
for all
α
>
1
\alpha > 1
α
>
1
.
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∫
1
4
(
4
x
3
−
x
2
+
3
x
+
2
)
d
x
\int_{1}^{4}\left(4 x^{3}-x^{2}+3 x+2\right) d x
∫
1
4
(
4
x
3
−
x
2
+
3
x
+
2
)
d
x
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1
x
+
s
∫
−
s
x
f
(
t
)
=
1
2
\frac{1}{x+s} \int_{-s}^{x} f(t)=\frac{1}{2}
x
+
s
1
∫
−
s
x
f
(
t
)
=
2
1
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What is the integral of
x
2
x^2
x
2
from
0
0
0
to
4
4
4
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∫
3
x
−
5
(
x
−
2
)
2
d
x
\int \frac{3 x-5}{(x-2)^{2}} d x
∫
(
x
−
2
)
2
3
x
−
5
d
x
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Find the value of
7
2
×
5
×
9
+
7
5
×
9
×
12
+
7
9
×
12
×
16
+
7
12
×
16
×
19
+
⋯
+
7
30
×
33
×
37
\frac{7}{2 \times 5 \times 9}+\frac{7}{5 \times 9 \times 12}+\frac{7}{9 \times 12 \times 16}+\frac{7}{12 \times 16 \times 19}+\cdots+\frac{7}{30 \times 33 \times 37}
2
×
5
×
9
7
+
5
×
9
×
12
7
+
9
×
12
×
16
7
+
12
×
16
×
19
7
+
⋯
+
30
×
33
×
37
7
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Does the series converge or diverge?
∑
n
=
1
∞
1
2
n
=
1
2
+
1
4
+
1
8
+
1
16
+
⋯
\sum_{n=1}^{\infty}\frac{1}{2^{n}}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots
n
=
1
∑
∞
2
n
1
=
2
1
+
4
1
+
8
1
+
16
1
+
⋯
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Evaluate the integral
∫
x
3
−
4
x
2
+
5
x
−
2
x
2
−
4
x
+
3
d
x
\int\frac{x^{3}-4x^{2}+5x-2}{x^{2}-4x+3}\,dx
∫
x
2
−
4
x
+
3
x
3
−
4
x
2
+
5
x
−
2
d
x
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The width of a rectangle measures
(
3
t
+
1
)
(3 t+1)
(
3
t
+
1
)
centimeters, and its length measures
(
10
t
+
5
)
(10 t+5)
(
10
t
+
5
)
centimeters. Which expression represents the perimeter, in centimeters, of the rectangle?
\newline
26
t
+
12
26 t+12
26
t
+
12
\newline
8
t
+
30
8 t+30
8
t
+
30
\newline
15
+
4
t
15+4 t
15
+
4
t
\newline
13
t
+
6
13 t+6
13
t
+
6
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∫
0
π
/
8
2
cos
4
t
sin
4
t
d
t
\int_{0}^{\pi / 8} 2^{\cos 4 t} \sin 4 t d t
∫
0
π
/8
2
c
o
s
4
t
sin
4
t
d
t
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∫
x
3
+
6
x
+
1
(
x
+
2
)
2
(
x
2
+
3
)
d
x
\int \frac{x^{3}+6 x+1}{(x+2)^{2}\left(x^{2}+3\right)} d x
∫
(
x
+
2
)
2
(
x
2
+
3
)
x
3
+
6
x
+
1
d
x
=
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Evaluate the integral.
∫
x
2
+
1
x
3
+
3
x
+
20
d
x
\int \frac{x^{2}+1}{\sqrt{x^{3}+3 x+20}} d x
∫
x
3
+
3
x
+
20
x
2
+
1
d
x
.
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Integrate
∫
t
2
+
1
(
t
2
+
2
t
+
3
)
2
d
t
\int \frac{t^{2}+1}{\left(t^{2}+2 t+3\right)^{2}} d t
∫
(
t
2
+
2
t
+
3
)
2
t
2
+
1
d
t
=
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19
−
38
y
=
76
x
24
x
=
−
6
(
2
y
−
1
)
\begin{aligned} 19-38y&=76x \newline24x&=-6(2y-1)\end{aligned}
19
−
38
y
24
x
=
76
x
=
−
6
(
2
y
−
1
)
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Evaluate the integral:
∫
x
5
−
x
4
−
3
x
+
5
x
4
−
2
x
3
+
2
x
2
−
2
x
+
1
\int \frac{x^{5}-x^{4}-3 x+5}{x^{4}-2 x^{3}+2 x^{2}-2 x+1}
∫
x
4
−
2
x
3
+
2
x
2
−
2
x
+
1
x
5
−
x
4
−
3
x
+
5
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Evaluate the expression
5
x
3
x
2
\frac{5 x^{3}}{x^{2}}
x
2
5
x
3
for
x
−
2
x-2
x
−
2
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If
∫
1
8
[
3
f
(
x
)
+
2
]
d
x
=
29
\int_{1}^{8}[3 f(x)+2] d x=29
∫
1
8
[
3
f
(
x
)
+
2
]
d
x
=
29
, find
∫
1
8
f
(
x
)
d
x
\int_{1}^{8} f(x) d x
∫
1
8
f
(
x
)
d
x
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Evaluate the integral:
∫
−
π
8
π
8
sin
2
(
2
x
)
d
x
\int_{\frac{-\pi}{8}}^{\frac{\pi}{8}} \sin ^{2}(2 x) d x
∫
8
−
π
8
π
sin
2
(
2
x
)
d
x
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Evaluate the integral:
∫
π
6
π
6
sin
2
(
2
x
)
d
x
\int_{\frac{\pi}{6}}^{\frac{\pi}{6}} \sin ^{2}(2 x) \mathrm{d} x
∫
6
π
6
π
sin
2
(
2
x
)
d
x
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Evaluate the integral:
∫
1
8
1
4
sin
2
(
2
x
)
d
x
\int_{\frac{1}{8}}^{\frac{1}{4}} \sin ^{2}(2 x) d x
∫
8
1
4
1
sin
2
(
2
x
)
d
x
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Evaluate the integral:
∫
π
8
π
8
sin
2
(
2
x
)
d
x
\int_{\frac{\pi}{8}}^{\frac{\pi}{8}} \sin ^{2}(2 x) d x
∫
8
π
8
π
sin
2
(
2
x
)
d
x
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Evaluate the integral:
∫
4
1
sin
2
(
2
x
)
d
x
\int_{4}^{1} \sin ^{2}(2 x) d x
∫
4
1
sin
2
(
2
x
)
d
x
Get tutor help
Evaluate the integral:
∫
0
2
d
x
x
2
+
4
\int_{0}^{2} \frac{d x}{\sqrt{x^{2}+4}}
∫
0
2
x
2
+
4
d
x
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Evaluate the limit:
∫
0
ln
3
e
−
x
1
+
e
−
x
d
x
\int_{0}^{\ln 3} \frac{\mathrm{e}^{-x}}{1+\mathrm{e}^{-x}} \mathrm{~d} x
∫
0
l
n
3
1
+
e
−
x
e
−
x
d
x
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what is the integral of
x
2
x^2
x
2
?
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lim
x
→
1
∫
1
x
e
t
2
x
2
−
1
=
\lim _{x \rightarrow 1} \frac{\int_{1}^{x} e^{t^{2}}}{x^{2}-1}=
lim
x
→
1
x
2
−
1
∫
1
x
e
t
2
=
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∫
1
2
x
x
+
4
d
x
=
1
+
4
ln
5
6
\int_{1}^{2} \frac{x}{x+4} d x=1+4 \ln \frac{5}{6}
∫
1
2
x
+
4
x
d
x
=
1
+
4
ln
6
5
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Evaluate
\newline
∫
−
3
4
(
2
k
x
+
3
k
x
2
)
d
x
\int_{-3}^{4}\left(2 k x+3 k x^{2}\right) d x
∫
−
3
4
(
2
k
x
+
3
k
x
2
)
d
x
\newline
giving your answer in terms of
k
k
k
.
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What is the sum of
(
w
6
+
2
3
w
2
)
\left(w^6+\dfrac{2}{3}w^2\right)
(
w
6
+
3
2
w
2
)
and
(
1
2
w
6
+
1
3
w
2
+
1
4
)
\left(\dfrac{1}{2}w^6+\dfrac{1}{3}w^2+\dfrac{1}{4}\right)
(
2
1
w
6
+
3
1
w
2
+
4
1
)
?
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∫
0
3
x
d
x
x
4
+
16
\int_{0}^{\sqrt{3}} \frac{x d x}{\sqrt{x^{4}+16}}
∫
0
3
x
4
+
16
x
d
x
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Evaluate
lim
x
→
10
x
2
−
99
x
−
10
\lim_{x \to 10}\frac{x^{2}-99}{x-10}
lim
x
→
10
x
−
10
x
2
−
99
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What is the product of
(
1
−
p
)
(1-p)
(
1
−
p
)
and
(
1
2
−
p
)
\left(\frac{1}{2}-p\right)
(
2
1
−
p
)
, all reduced by
P
P
P
?
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Write an integral expression that will give the length of the path given by
f
(
x
)
=
10
x
4
−
1
f(x)=10x^{4}-1
f
(
x
)
=
10
x
4
−
1
from
x
=
7
x=7
x
=
7
to
x
=
8
x=8
x
=
8
.
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Evaluate and write the answer in scientific notation:
\newline
(
1.48
×
1
0
3
)
(
7
×
1
0
4
)
÷
8.2
×
1
0
12
(1.48\times10^{3})(7\times10^{4})\div8.2\times10^{12}
(
1.48
×
1
0
3
)
(
7
×
1
0
4
)
÷
8.2
×
1
0
12
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Find
d
d
t
∫
2
t
4
e
x
2
d
x
\frac{d}{dt}\int_{2}^{t^{4}}e^{x^{2}}dx
d
t
d
∫
2
t
4
e
x
2
d
x
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∫
1
1
−
sin
x
d
x
\int \frac{1}{1-\sin x} \, dx
∫
1
−
s
i
n
x
1
d
x
Get tutor help
∫
2
x
−
3
9
+
x
2
d
x
\int\frac{2x-3}{9+x^{2}}\,dx
∫
9
+
x
2
2
x
−
3
d
x
Get tutor help
For the polynomial function
p
(
x
)
=
7
x
2
−
3
x
−
10
p(x)=7x^{2}-3x-10
p
(
x
)
=
7
x
2
−
3
x
−
10
, evaluate
p
(
4
)
p(4)
p
(
4
)
.
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∫
0
1
x
ln
x
d
x
\int_{0}^{1}x \ln x\,dx
∫
0
1
x
ln
x
d
x
Get tutor help
Evaluate
∫
1
3
7
ln
x
x
d
x
\int_{1}^{3} \frac{7^{\ln x}}{x} \, dx
∫
1
3
x
7
l
n
x
d
x
Get tutor help
Evaluate
∑
n
=
1
∞
(
sin
n
2
n
)
\sum_{n=1}^{\infty}\left(\frac{\sin n}{2^{n}}\right)
∑
n
=
1
∞
(
2
n
s
i
n
n
)
Get tutor help
Solve:
f
(
x
)
=
∫
0
x
e
−
t
2
d
t
f(x) = \int_{0}^{x} e^{-t^{2}} \, dt
f
(
x
)
=
∫
0
x
e
−
t
2
d
t
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Solve:
∫
x
3
+
1
x
3
−
5
x
2
+
6
x
d
x
\int\frac{x^{3}+1}{x^{3}-5x^{2}+6x}\,dx
∫
x
3
−
5
x
2
+
6
x
x
3
+
1
d
x
.
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