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Math Problems
Calculus
Find indefinite integrals using the substitution
Factor the expression completely.
\newline
x
y
+
x
3
y
5
x y+x^{3} y^{5}
x
y
+
x
3
y
5
\newline
Answer:
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Calculate the integral and write your answer in simplest form.
\newline
∫
3
x
5
2
d
x
\int \frac{3 \sqrt{x^{5}}}{2} \mathrm{dx}
∫
2
3
x
5
dx
\newline
Answer:
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Calculate the integral and write your answer in simplest form.
\newline
∫
5
x
3
4
d
x
\int \frac{5 \sqrt{x^{3}}}{4} \mathrm{dx}
∫
4
5
x
3
dx
\newline
Answer:
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Calculate the integral and write your answer in simplest form.
\newline
∫
x
5
4
d
x
\int \frac{\sqrt{x^{5}}}{4} \mathrm{dx}
∫
4
x
5
dx
\newline
Answer:
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What is the equation of the line that passes through the point
(
5
,
6
)
(5,6)
(
5
,
6
)
and has a slope of
−
2
5
-\frac{2}{5}
−
5
2
?
\newline
Answer:
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A function is invertible if it is
\newline
a) surjective
\newline
b) bijective
\newline
c) injective
\newline
d) neither surjective nor injective
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∫
x
3
−
2
x
2
+
1
(
x
2
−
2
x
+
5
)
2
d
x
\int\frac{x^{3}-2x^{2}+1}{(x^{2}-2x+5)^{2}}dx
∫
(
x
2
−
2
x
+
5
)
2
x
3
−
2
x
2
+
1
d
x
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∫
cos
(
12
r
)
cos
(
15
r
)
d
r
\int \cos(12r) \cos(15r) \, dr
∫
cos
(
12
r
)
cos
(
15
r
)
d
r
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∫
1
x
2
d
x
\int \frac{1}{x^{2}} \, dx
∫
x
2
1
d
x
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Integrate
∫
x
3
+
x
x
2
+
2
d
x
\int\frac{x^{3}+x}{x^{2}+2}\,dx
∫
x
2
+
2
x
3
+
x
d
x
.
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∫
5
5
x
d
x
\int 5\sqrt{5x}\,dx
∫
5
5
x
d
x
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∫
(
x
−
3
)
d
x
x
+
3
\int \frac{(x-3)\,dx}{x+3}
∫
x
+
3
(
x
−
3
)
d
x
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Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).
\newline
lim
x
→
∞
−
50
x
6
+
4
+
27
x
12
3
x
4
+
4
x
2
\lim _{x \rightarrow \infty} \frac{\sqrt[3]{-50 x^{6}+4+27 x^{12}}}{x^{4}+4 x^{2}}
x
→
∞
lim
x
4
+
4
x
2
3
−
50
x
6
+
4
+
27
x
12
\newline
Answer:
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Let
f
f
f
be the function defined by
f
(
x
)
=
x
2
f(x)=x^{2}
f
(
x
)
=
x
2
. If four subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
2
3
x
2
d
x
\int_{2}^{3} x^{2} d x
∫
2
3
x
2
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
)
=
6
ln
(
x
)
f(x)=6 \ln (x)
f
(
x
)
=
6
ln
(
x
)
. If four subintervals of equal length are used, what is the value of the left Riemann sum approximation for
∫
1
9
6
ln
(
x
)
d
x
\int_{1}^{9} 6 \ln (x) d x
∫
1
9
6
ln
(
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
)
=
4
x
f(x)=4 \sqrt{x}
f
(
x
)
=
4
x
. If four subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
1
7
4
x
d
x
\int_{1}^{7} 4 \sqrt{x} d x
∫
1
7
4
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
)
=
4
x
f(x)=4 \sqrt{x}
f
(
x
)
=
4
x
. If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
3
4.5
4
x
d
x
\int_{3}^{4.5} 4 \sqrt{x} d x
∫
3
4.5
4
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
)
=
3
ln
(
x
)
f(x)=3 \ln (x)
f
(
x
)
=
3
ln
(
x
)
. If three subintervals of equal length are used, what is the value of the left Riemann sum approximation for
∫
1
10
3
ln
(
x
)
d
x
\int_{1}^{10} 3 \ln (x) d x
∫
1
10
3
ln
(
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
)
=
4
x
f(x)=4 \sqrt{x}
f
(
x
)
=
4
x
. If four subintervals of equal length are used, what is the value of the midpoint Riemann sum approximation for
∫
2
8
4
x
d
x
\int_{2}^{8} 4 \sqrt{x} d x
∫
2
8
4
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
)
=
5
ln
(
x
)
f(x)=5 \ln (x)
f
(
x
)
=
5
ln
(
x
)
. If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
3
7.5
5
ln
(
x
)
d
x
?
\int_{3}^{7.5} 5 \ln (x) d x ?
∫
3
7.5
5
ln
(
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
)
=
x
4
f(x)=x^{4}
f
(
x
)
=
x
4
. If five subintervals of equal length are used, what is the value of the midpoint Riemann sum approximation for
∫
1
2
x
4
d
x
\int_{1}^{2} x^{4} d x
∫
1
2
x
4
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
)
=
4
x
f(x)=4 \sqrt{x}
f
(
x
)
=
4
x
. If three subintervals of equal length are used, what is the value of the right Riemann sum approximation for
∫
0
9
4
x
d
x
\int_{0}^{9} 4 \sqrt{x} d x
∫
0
9
4
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
)
=
3
ln
(
x
)
f(x)=3 \ln (x)
f
(
x
)
=
3
ln
(
x
)
. If six subintervals of equal length are used, what is the value of the left Riemann sum approximation for
∫
1
4
3
ln
(
x
)
d
x
\int_{1}^{4} 3 \ln (x) d x
∫
1
4
3
ln
(
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
)
=
3
ln
(
x
)
f(x)=3 \ln (x)
f
(
x
)
=
3
ln
(
x
)
. If three subintervals of equal length are used, what is the value of the trapezoidal sum approximation for
∫
2
3.5
3
ln
(
x
)
d
x
\int_{2}^{3.5} 3 \ln (x) d x
∫
2
3.5
3
ln
(
x
)
d
x
? Round to the nearest thousandth if necessary.
\newline
Answer:
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Find the value of
∫
1
5
6
d
x
11
−
2
x
\int_{1}^{5} \frac{6 d x}{11-2 x}
∫
1
5
11
−
2
x
6
d
x
. Express your answer as a constant times
ln
3
\ln 3
ln
3
.
\newline
Answer:
□
ln
3
\square \ln 3
□
ln
3
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Find the value of
∫
3
5
3
d
x
7
−
x
\int_{3}^{5} \frac{3 d x}{7-x}
∫
3
5
7
−
x
3
d
x
. Express your answer as a constant times
ln
2
\ln 2
ln
2
.
\newline
Answer:
□
ln
2
\square \ln 2
□
ln
2
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Find the value of
∫
2
7
5
d
x
x
−
12
\int_{2}^{7} \frac{5 d x}{x-12}
∫
2
7
x
−
12
5
d
x
. Express your answer as a constant times
ln
2
\ln 2
ln
2
.
\newline
Answer:
□
ln
2
\square \ln 2
□
ln
2
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Find the value of
∫
4
6
9
10
−
3
x
d
x
\int_{4}^{6} \frac{9}{10-3 x} d x
∫
4
6
10
−
3
x
9
d
x
. Write your answer as the logarithm of a single number in simplest form.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
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Find the value of
∫
4
6
3
d
x
x
−
8
\int_{4}^{6} \frac{3 d x}{x-8}
∫
4
6
x
−
8
3
d
x
. Express your answer as a constant times
ln
2
\ln 2
ln
2
.
\newline
Answer:
□
ln
2
\square \ln 2
□
ln
2
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Find the value of
∫
4
9
1
3
x
−
11
d
x
\int_{4}^{9} \frac{1}{3 x-11} d x
∫
4
9
3
x
−
11
1
d
x
. Express your answer as a constant times
ln
2
\ln 2
ln
2
.
\newline
Answer:
□
ln
2
\square \ln 2
□
ln
2
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Find the value of
∫
7
9
2
d
x
x
−
11
\int_{7}^{9} \frac{2 d x}{x-11}
∫
7
9
x
−
11
2
d
x
. Write your answer as the logarithm of a single number in simplest form.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
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Find the value of
∫
1
4
4
2
x
−
9
d
x
\int_{1}^{4} \frac{4}{2 x-9} d x
∫
1
4
2
x
−
9
4
d
x
. Express your answer as a constant times
ln
7
\ln 7
ln
7
.
\newline
Answer:
□
ln
7
\square \ln 7
□
ln
7
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Find the value of
∫
6
8
3
x
−
4
d
x
\int_{6}^{8} \frac{3}{x-4} d x
∫
6
8
x
−
4
3
d
x
. Write your answer as the logarithm of a single number in simplest form.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
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Find the value of
∫
4
6
5
8
−
x
d
x
\int_{4}^{6} \frac{5}{8-x} d x
∫
4
6
8
−
x
5
d
x
. Write your answer as the logarithm of a single number in simplest form.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
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Find the value of
∫
6
8
4
d
x
4
−
x
\int_{6}^{8} \frac{4 d x}{4-x}
∫
6
8
4
−
x
4
d
x
. Express your answer as a constant times
ln
2
\ln 2
ln
2
.
\newline
Answer:
□
ln
2
\square \ln 2
□
ln
2
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Find the value of
∫
6
8
2
d
x
4
−
x
\int_{6}^{8} \frac{2 d x}{4-x}
∫
6
8
4
−
x
2
d
x
. Write your answer as the logarithm of a single number in simplest form.
\newline
Answer:
ln
(
□
)
\ln (\square)
ln
(
□
)
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Find the value of
∫
3
8
3
d
x
9
−
x
\int_{3}^{8} \frac{3 d x}{9-x}
∫
3
8
9
−
x
3
d
x
. Express your answer as a constant times
ln
6
\ln 6
ln
6
.
\newline
Answer:
□
ln
6
\square \ln 6
□
ln
6
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Evaluate the integral and express your answer in simplest form.
\newline
∫
−
9
x
x
2
−
25
d
x
\int \frac{-9}{x \sqrt{x^{2}-25}} d x
∫
x
x
2
−
25
−
9
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
−
3
x
x
2
−
1
d
x
\int \frac{-3}{x \sqrt{x^{2}-1}} d x
∫
x
x
2
−
1
−
3
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
9
x
x
2
−
1
d
x
\int \frac{9}{x \sqrt{x^{2}-1}} d x
∫
x
x
2
−
1
9
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
−
7
16
−
4
x
2
d
x
\int \frac{-7}{\sqrt{16-4 x^{2}}} d x
∫
16
−
4
x
2
−
7
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
2
36
−
16
x
2
d
x
\int \frac{2}{\sqrt{36-16 x^{2}}} d x
∫
36
−
16
x
2
2
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
−
3
4
−
16
x
2
d
x
\int \frac{-3}{\sqrt{4-16 x^{2}}} d x
∫
4
−
16
x
2
−
3
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
4
1
−
x
2
d
x
\int \frac{4}{\sqrt{1-x^{2}}} d x
∫
1
−
x
2
4
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
3
x
x
2
−
9
d
x
\int \frac{3}{x \sqrt{x^{2}-9}} d x
∫
x
x
2
−
9
3
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
10
16
−
4
x
2
d
x
\int \frac{10}{\sqrt{16-4 x^{2}}} d x
∫
16
−
4
x
2
10
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
−
2
x
x
2
−
9
d
x
\int \frac{-2}{x \sqrt{x^{2}-9}} d x
∫
x
x
2
−
9
−
2
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
−
1
25
−
9
x
2
d
x
\int \frac{-1}{\sqrt{25-9 x^{2}}} d x
∫
25
−
9
x
2
−
1
d
x
\newline
Answer:
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Evaluate the integral and express your answer in simplest form.
\newline
∫
6
4
−
x
2
d
x
\int \frac{6}{\sqrt{4-x^{2}}} d x
∫
4
−
x
2
6
d
x
\newline
Answer:
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Evaluate
∫
0
4
(
3
e
−
0.25
x
+
8
)
d
x
\int_{0}^{4}\left(3 e^{-0.25 x}+8\right) d x
∫
0
4
(
3
e
−
0.25
x
+
8
)
d
x
and express the answer in simplest form.
\newline
Answer:
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