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Math Problems
Calculus
Evaluate definite integrals using the chain rule
i)
∫
0
1
x
e
2
x
cos
3
(
x
)
d
x
\int_{0}^{1} x e^{2x} \cos^{3}(x) \, dx
∫
0
1
x
e
2
x
cos
3
(
x
)
d
x
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∫
−
1
1
1
2
−
x
d
x
\int_{-1}^{1}\frac{1}{2-x}dx
∫
−
1
1
2
−
x
1
d
x
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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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∫
6
x
−
11
(
x
−
1
)
2
d
x
\int\frac{6x-11}{(x-1)^{2}}dx
∫
(
x
−
1
)
2
6
x
−
11
d
x
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∫
9
x
+
2
x
2
+
x
−
6
d
x
\int\frac{9x+2}{x^{2}+x-6}\,dx
∫
x
2
+
x
−
6
9
x
+
2
d
x
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Evaluate
∫
x
2
+
2
x
+
3
x
+
1
d
x
\int\frac{x^{2}+2x+3}{x+1}\,dx
∫
x
+
1
x
2
+
2
x
+
3
d
x
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∫
1
e
e
tan
−
1
x
x
d
x
\int_{\frac{1}{e}}^{e} \frac{\tan^{-1}x}{x}dx
∫
e
1
e
x
t
a
n
−
1
x
d
x
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∫
0
π
sin
(
3
x
+
π
2
)
d
x
\int_{0}^{\pi}\sin(3x+\frac{\pi}{2})dx
∫
0
π
sin
(
3
x
+
2
π
)
d
x
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∫
10
x
3
−
5
x
x
4
−
x
2
+
6
d
x
\int\frac{10x^{3}-5x}{\sqrt{x^{4}-x^{2}+6}}\,dx
∫
x
4
−
x
2
+
6
10
x
3
−
5
x
d
x
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∫
e
−
x
+
sin
x
e
−
x
+
cos
x
d
x
\int \frac{e^{-x} + \sin x}{e^{-x} + \cos x} \, dx
∫
e
−
x
+
c
o
s
x
e
−
x
+
s
i
n
x
d
x
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∫
e
−
x
+
sin
x
cos
x
+
e
−
x
d
x
\int\frac{e^{-x}+\sin x}{\cos x+e^{-x}}\,dx
∫
c
o
s
x
+
e
−
x
e
−
x
+
s
i
n
x
d
x
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c)
∫
0
1
x
e
2
x
cos
3
(
x
)
d
x
\int_{0}^{1} x e^{2x} \cos^{3}(x) \, dx
∫
0
1
x
e
2
x
cos
3
(
x
)
d
x
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lim
x
→
0
+
(
e
x
2
−
cosh
(
x
)
(
x
+
x
5
)
α
)
\lim_{x \to 0^{+}}\left(\frac{e^{\frac{x}{2}}-\cosh(\sqrt{x})}{(x+\sqrt[5]{x})^{\alpha}}\right)
x
→
0
+
lim
(
(
x
+
5
x
)
α
e
2
x
−
cosh
(
x
)
)
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∫
x
3
d
x
(
x
2
+
1
)
3
2
\int \frac{x^{3} \, dx}{(x^{2}+1)^{\frac{3}{2}}}
∫
(
x
2
+
1
)
2
3
x
3
d
x
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a)
∫
0
π
2
x
sin
x
d
x
\int_{0}^{\frac{\pi}{2}} x \sin x \, dx
∫
0
2
π
x
sin
x
d
x
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Let
f
f
f
be the function defined by
f
(
x
)
=
2
x
f(x)=2^{x}
f
(
x
)
=
2
x
. If six subintervals of equal length are used, what is the value of the right Riemann sum approximation for
∫
2
3.5
2
x
d
x
\int_{2}^{3.5} 2^{x} d x
∫
2
3.5
2
x
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
2
x
f(x)=2^{x}
f
(
x
)
=
2
x
. If three subintervals of equal length are used, what is the value of the right Riemann sum approximation for
∫
1
2.5
2
x
d
x
\int_{1}^{2.5} 2^{x} d x
∫
1
2.5
2
x
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Evaluate
∫
0
20
(
9
e
0.2
x
+
2
)
d
x
\int_{0}^{20}\left(9 e^{0.2 x}+2\right) d x
∫
0
20
(
9
e
0.2
x
+
2
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
4
(
2
e
−
0.25
x
−
4
x
)
d
x
\int_{0}^{4}\left(2 e^{-0.25 x}-4 x\right) d x
∫
0
4
(
2
e
−
0.25
x
−
4
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
5
(
7
e
−
0.2
x
−
4
x
)
d
x
\int_{0}^{5}\left(7 e^{-0.2 x}-4 x\right) d x
∫
0
5
(
7
e
−
0.2
x
−
4
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
5
(
3
e
−
0.2
x
−
2
x
)
d
x
\int_{0}^{5}\left(3 e^{-0.2 x}-2 x\right) d x
∫
0
5
(
3
e
−
0.2
x
−
2
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
10
(
3
e
0.2
x
+
4
)
d
x
\int_{0}^{10}\left(3 e^{0.2 x}+4\right) d x
∫
0
10
(
3
e
0.2
x
+
4
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
4
(
5
e
−
0.25
x
−
4
x
)
d
x
\int_{0}^{4}\left(5 e^{-0.25 x}-4 x\right) d x
∫
0
4
(
5
e
−
0.25
x
−
4
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
2
(
2
e
0.5
x
−
2
x
)
d
x
\int_{0}^{2}\left(2 e^{0.5 x}-2 x\right) d x
∫
0
2
(
2
e
0.5
x
−
2
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
4
(
9
e
0.25
x
+
4
x
)
d
x
\int_{0}^{4}\left(9 e^{0.25 x}+4 x\right) d x
∫
0
4
(
9
e
0.25
x
+
4
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
6
(
10
e
−
0.5
x
−
2
x
)
d
x
\int_{0}^{6}\left(10 e^{-0.5 x}-2 x\right) d x
∫
0
6
(
10
e
−
0.5
x
−
2
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
5
(
7
e
0.2
x
+
4
x
)
d
x
\int_{0}^{5}\left(7 e^{0.2 x}+4 x\right) d x
∫
0
5
(
7
e
0.2
x
+
4
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
30
(
2
e
−
0.2
x
−
1
)
d
x
\int_{0}^{30}\left(2 e^{-0.2 x}-1\right) d x
∫
0
30
(
2
e
−
0.2
x
−
1
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
2
(
10
e
−
0.5
x
+
2
x
)
d
x
\int_{0}^{2}\left(10 e^{-0.5 x}+2 x\right) d x
∫
0
2
(
10
e
−
0.5
x
+
2
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
5
(
3
e
0.2
x
−
2
x
)
d
x
\int_{0}^{5}\left(3 e^{0.2 x}-2 x\right) d x
∫
0
5
(
3
e
0.2
x
−
2
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
2
(
9
e
−
0.5
x
−
4
x
)
d
x
\int_{0}^{2}\left(9 e^{-0.5 x}-4 x\right) d x
∫
0
2
(
9
e
−
0.5
x
−
4
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
15
(
9
e
−
0.2
x
−
1
)
d
x
\int_{0}^{15}\left(9 e^{-0.2 x}-1\right) d x
∫
0
15
(
9
e
−
0.2
x
−
1
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
4
(
8
e
−
0.5
x
−
2
x
)
d
x
\int_{0}^{4}\left(8 e^{-0.5 x}-2 x\right) d x
∫
0
4
(
8
e
−
0.5
x
−
2
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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Evaluate
∫
0
4
(
9
e
0.5
x
−
2
x
)
d
x
\int_{0}^{4}\left(9 e^{0.5 x}-2 x\right) d x
∫
0
4
(
9
e
0.5
x
−
2
x
)
d
x
and express the answer in simplest form.
\newline
Answer:
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The value of
\newline
∫
1
3
d
x
1
+
x
2
\int_{1}^{\sqrt{3}}\frac{dx}{1+x^{2}}
∫
1
3
1
+
x
2
d
x
is:
\newline
(a)
π
2
\frac{\pi}{2}
2
π
\newline
(b)
2
π
3
\frac{2\pi}{3}
3
2
π
\newline
(c)
π
6
\frac{\pi}{6}
6
π
\newline
(d)
π
12
\frac{\pi}{12}
12
π
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∑
n
=
1
∞
n
2
+
1
5
n
\sum_{n=1}^{\infty}\frac{n^{2}+1}{5^{n}}
n
=
1
∑
∞
5
n
n
2
+
1
Get tutor help
∫
1
2
2
2
x
4
x
5
+
1
d
x
\int_{\frac{1}{2}}^{2}\frac{2x^{4}}{x^{5}+1}\,dx
∫
2
1
2
x
5
+
1
2
x
4
d
x
Get tutor help
∫
[
(
x
−
1
)
5
+
3
(
x
−
1
)
2
+
5
]
d
x
\int[(x-1)^{5}+3(x-1)^{2}+5]\,dx
∫
[(
x
−
1
)
5
+
3
(
x
−
1
)
2
+
5
]
d
x
Get tutor help
∫
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
Get tutor help
∫
1
(
2
x
+
1
)
x
2
+
2
x
+
2
d
x
\int \frac{1}{(2x+1)\sqrt{x^{2}+2x+2}}\,dx
∫
(
2
x
+
1
)
x
2
+
2
x
+
2
1
d
x
Get tutor help
Evaluate
∫
11
e
3
+
10
2
x
−
19
x
−
10
d
x
\int_{11}^{e^{3}+10} \frac{2 x-19}{x-10} d x
∫
11
e
3
+
10
x
−
10
2
x
−
19
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
3
e
3
+
2
2
x
−
1
x
−
2
d
x
\int_{3}^{e^{3}+2} \frac{2 x-1}{x-2} d x
∫
3
e
3
+
2
x
−
2
2
x
−
1
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
2
3
4
x
2
−
11
x
−
22
x
−
4
d
x
\int_{2}^{3} \frac{4 x^{2}-11 x-22}{x-4} d x
∫
2
3
x
−
4
4
x
2
−
11
x
−
22
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
2
3
4
x
2
−
11
x
−
22
x
−
4
d
x
\int_{2}^{3} \frac{4 x^{2}-11 x-22}{x-4} d x
∫
2
3
x
−
4
4
x
2
−
11
x
−
22
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
\newline
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Evaluate
∫
1
4
4
x
2
−
27
x
−
5
x
−
7
d
x
\int_{1}^{4} \frac{4 x^{2}-27 x-5}{x-7} d x
∫
1
4
x
−
7
4
x
2
−
27
x
−
5
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
3
7
4
x
2
−
5
x
−
12
x
+
1
d
x
\int_{3}^{7} \frac{4 x^{2}-5 x-12}{x+1} d x
∫
3
7
x
+
1
4
x
2
−
5
x
−
12
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
2
−
3
x
+
2
x
+
3
d
x
\int_{-2}^{e^{2}-3} \frac{x+2}{x+3} d x
∫
−
2
e
2
−
3
x
+
3
x
+
2
d
x
. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).
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Evaluate
∫
−
8
e
−
9
3
x
+
25
x
+
9
d
x
\int_{-8}^{e-9} \frac{3 x+25}{x+9} d x
∫
−
8
e
−
9
x
+
9
3
x
+
25
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
−
43
x
−
10
d
x
\int_{11}^{e^{2}+10} \frac{4 x-43}{x-10} d x
∫
11
e
2
+
10
x
−
10
4
x
−
43
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).
\newline
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