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Math Problems
Calculus
Evaluate definite integrals using the chain rule
Evaluate
∫
6
e
3
+
5
3
x
−
16
x
−
5
d
x
\int_{6}^{e^{3}+5} \frac{3 x-16}{x-5} d x
∫
6
e
3
+
5
x
−
5
3
x
−
16
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
1
8
x
−
8
x
−
9
d
x
\int_{1}^{8} \frac{x-8}{x-9} d x
∫
1
8
x
−
9
x
−
8
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
1
2
4
x
2
−
21
x
+
30
x
−
3
d
x
\int_{1}^{2} \frac{4 x^{2}-21 x+30}{x-3} d x
∫
1
2
x
−
3
4
x
2
−
21
x
+
30
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
6
12
x
−
7
x
−
5
d
x
\int_{6}^{12} \frac{x-7}{x-5} d x
∫
6
12
x
−
5
x
−
7
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
1
3
4
x
2
−
19
x
+
9
x
−
4
d
x
\int_{1}^{3} \frac{4 x^{2}-19 x+9}{x-4} d x
∫
1
3
x
−
4
4
x
2
−
19
x
+
9
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
6
e
2
+
5
4
x
−
21
x
−
5
d
x
\int_{6}^{e^{2}+5} \frac{4 x-21}{x-5} d x
∫
6
e
2
+
5
x
−
5
4
x
−
21
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
11
e
2
+
10
4
x
−
41
x
−
10
d
x
\int_{11}^{e^{2}+10} \frac{4 x-41}{x-10} d x
∫
11
e
2
+
10
x
−
10
4
x
−
41
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
2
e
+
1
4
x
−
3
x
−
1
d
x
\int_{2}^{e+1} \frac{4 x-3}{x-1} d x
∫
2
e
+
1
x
−
1
4
x
−
3
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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∫
0
π
20
20
tan
5
x
d
x
\int_{0}^{\frac{\pi}{20}} 20 \tan 5x \, dx
∫
0
20
π
20
tan
5
x
d
x
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∫
x
+
1
+
2
(
x
+
1
)
2
−
x
+
1
d
x
\int\frac{\sqrt{x+1}+2}{(x+1)^{2}-\sqrt{x+1}}dx
∫
(
x
+
1
)
2
−
x
+
1
x
+
1
+
2
d
x
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Evaluate the integral.
\newline
∫
(
x
2
+
5
)
2
d
x
\int\left(x^{2}+5\right)^{2} \mathrm{~d} x
∫
(
x
2
+
5
)
2
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
x
(
3
x
+
4
)
2
d
x
\int x(3 x+4)^{2} \mathrm{~d} x
∫
x
(
3
x
+
4
)
2
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
x
2
(
3
x
−
4
)
(
x
−
3
)
d
x
\int x^{2}(3 x-4)(x-3) \mathrm{d} x
∫
x
2
(
3
x
−
4
)
(
x
−
3
)
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
x
2
(
4
x
−
5
)
2
d
x
\int x^{2}(4 x-5)^{2} \mathrm{~d} x
∫
x
2
(
4
x
−
5
)
2
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
(
3
x
2
+
2
)
2
d
x
\int\left(3 x^{2}+2\right)^{2} \mathrm{~d} x
∫
(
3
x
2
+
2
)
2
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
(
3
x
2
−
5
)
2
d
x
\int\left(3 x^{2}-5\right)^{2} \mathrm{~d} x
∫
(
3
x
2
−
5
)
2
d
x
\newline
Answer:
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∫
0
1
x
5
e
1
−
x
6
d
x
\int_{0}^{1} x^{5} e^{1-x^{6}} \, dx
∫
0
1
x
5
e
1
−
x
6
d
x
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∫
x
tan
−
1
(
x
)
(
1
+
x
2
)
3
2
d
x
\int \frac{x\tan^{-1}(x)}{(1+x^{2})^{\frac{3}{2}}}dx
∫
(
1
+
x
2
)
2
3
x
t
a
n
−
1
(
x
)
d
x
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∫
−
1
1
+
x
2
×
1
1
+
x
2
d
x
\int \frac{-1}{\sqrt{1+x^{2}}} \times \frac{1}{1+x^{2}} \, dx
∫
1
+
x
2
−
1
×
1
+
x
2
1
d
x
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Find
∫
d
x
a
x
+
b
\int \frac{dx}{ax+b}
∫
a
x
+
b
d
x
\newline
(
1
1
1
)
log
e
(
a
x
+
b
)
+
c
\log_{e}(ax+b)+c
lo
g
e
(
a
x
+
b
)
+
c
\newline
(
3
3
3
)
C
+
1
a
log
e
(
a
x
+
b
)
C+\frac{1}{a}\log_{e}(ax+b)
C
+
a
1
lo
g
e
(
a
x
+
b
)
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∫
(
−
∞
)
(
∞
)
1
1
+
x
2
d
x
\int_{(-\infty)}^{(\infty)}\frac{1}{1+x^{2}}dx
∫
(
−
∞
)
(
∞
)
1
+
x
2
1
d
x
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∫
6
x
2
3
x
+
5
d
x
\int\frac{6x^{2}}{\sqrt{3x+5}}dx
∫
3
x
+
5
6
x
2
d
x
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∫
1
2
1
+
2
ln
(
x
)
(
1
+
ln
(
x
)
)
2
x
d
x
\int_{1}^{2}\frac{1+2\ln(x)}{(1+\ln(x))^{2}x}\,dx
∫
1
2
(
1
+
ln
(
x
)
)
2
x
1
+
2
ln
(
x
)
d
x
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∫
1
2
x
⋅
2
x
2
d
x
\int_{1}^{\sqrt{2}} x \cdot 2^{x^{2}} \, dx
∫
1
2
x
⋅
2
x
2
d
x
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∫
1
cos
(
x
)
2
d
x
\int \frac{1}{\cos(x)^{2}}\,dx
∫
c
o
s
(
x
)
2
1
d
x
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∫
d
x
−
2
x
2
+
8
x
+
4
\int \frac{dx}{\sqrt{-2x^{2}+8x+4}}
∫
−
2
x
2
+
8
x
+
4
d
x
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∫
0
2
d
x
x
2
−
2
x
+
2
\int_{0}^{2}\frac{dx}{x^{2}-2x+2}
∫
0
2
x
2
−
2
x
+
2
d
x
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∑
n
=
1
∞
3
n
n
2
n
!
\sum_{n=1}^{\infty}\frac{3^{n}n^{2}}{n!}
n
=
1
∑
∞
n
!
3
n
n
2
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f
)
lim
n
→
+
∞
[
2
n
+
(
3
5
)
n
]
f) \lim_{n \to +\infty}\left[\frac{2}{n}+\left(\frac{3}{5}\right)^n\right]
f
)
lim
n
→
+
∞
[
n
2
+
(
5
3
)
n
]
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Evaluate the integral.
\newline
∫
−
4
x
2
e
4
x
d
x
\int-4 x^{2} e^{4 x} d x
∫
−
4
x
2
e
4
x
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
−
2
x
3
cos
(
3
x
)
d
x
\int-2 x^{3} \cos (3 x) d x
∫
−
2
x
3
cos
(
3
x
)
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
−
4
x
2
3
3
x
d
x
\int-4 x^{2} 3^{3 x} d x
∫
−
4
x
2
3
3
x
d
x
\newline
Answer:
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The base of a solid is the region enclosed by the graphs of
y
=
sin
(
x
)
y=\sin (x)
y
=
sin
(
x
)
and
y
=
4
−
x
y=4-\sqrt{x}
y
=
4
−
x
, between
x
=
2
x=2
x
=
2
and
x
=
7
x=7
x
=
7
.
\newline
Cross sections of the solid perpendicular to the
x
x
x
-axis are rectangles whose height is
2
x
2 x
2
x
.
\newline
Which one of the definite integrals gives the volume of the solid?
\newline
Choose
1
1
1
answer:
\newline
(A)
∫
2
7
[
(
4
−
x
)
2
−
sin
2
(
x
)
]
d
x
\int_{2}^{7}\left[(4-\sqrt{x})^{2}-\sin ^{2}(x)\right] d x
∫
2
7
[
(
4
−
x
)
2
−
sin
2
(
x
)
]
d
x
\newline
(B)
∫
2
7
[
sin
(
x
)
+
x
−
4
]
⋅
2
x
d
x
\int_{2}^{7}[\sin (x)+\sqrt{x}-4] \cdot 2 x d x
∫
2
7
[
sin
(
x
)
+
x
−
4
]
⋅
2
x
d
x
\newline
(C)
∫
2
7
[
sin
2
(
x
)
−
(
4
−
x
)
2
]
d
x
\int_{2}^{7}\left[\sin ^{2}(x)-(4-\sqrt{x})^{2}\right] d x
∫
2
7
[
sin
2
(
x
)
−
(
4
−
x
)
2
]
d
x
\newline
(D)
∫
2
7
[
4
−
x
−
sin
(
x
)
]
⋅
2
x
d
x
\int_{2}^{7}[4-\sqrt{x}-\sin (x)] \cdot 2 x d x
∫
2
7
[
4
−
x
−
sin
(
x
)]
⋅
2
x
d
x
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The base of a solid is the region enclosed by the graphs of
y
=
sin
(
x
)
y=\sin (x)
y
=
sin
(
x
)
and
y
=
4
−
x
y=4-\sqrt{x}
y
=
4
−
x
, between
x
=
2
x=2
x
=
2
and
x
=
7
x=7
x
=
7
.
\newline
Cross sections of the solid perpendicular to the
x
x
x
-axis are rectangles whose height is
2
x
2 x
2
x
.
\newline
Which one of the definite integrals gives the volume of the solid?
\newline
Choose
1
1
1
answer:
\newline
(A)
∫
2
7
[
sin
(
x
)
+
x
−
4
]
⋅
2
x
d
x
\int_{2}^{7}[\sin (x)+\sqrt{x}-4] \cdot 2 x d x
∫
2
7
[
sin
(
x
)
+
x
−
4
]
⋅
2
x
d
x
\newline
(B)
∫
2
7
[
sin
2
(
x
)
−
(
4
−
x
)
2
]
d
x
\int_{2}^{7}\left[\sin ^{2}(x)-(4-\sqrt{x})^{2}\right] d x
∫
2
7
[
sin
2
(
x
)
−
(
4
−
x
)
2
]
d
x
\newline
(C)
∫
2
7
[
4
−
x
−
sin
(
x
)
]
⋅
2
x
d
x
\int_{2}^{7}[4-\sqrt{x}-\sin (x)] \cdot 2 x d x
∫
2
7
[
4
−
x
−
sin
(
x
)]
⋅
2
x
d
x
\newline
(D)
∫
2
7
[
(
4
−
x
)
2
−
sin
2
(
x
)
]
d
x
\int_{2}^{7}\left[(4-\sqrt{x})^{2}-\sin ^{2}(x)\right] d x
∫
2
7
[
(
4
−
x
)
2
−
sin
2
(
x
)
]
d
x
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