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Math Problems
Calculus
Find indefinite integrals using the substitution and by parts
The function
g
(
x
)
g(x)
g
(
x
)
is odd and continuous for all
x
\mathrm{x}
x
. If
∫
0
a
g
(
x
)
d
x
=
3.5
\int_{0}^{a} g(x) d x=3.5
∫
0
a
g
(
x
)
d
x
=
3.5
, what is
∫
−
a
a
g
(
x
)
d
x
?
\int_{-a}^{a} g(x) d x ?
∫
−
a
a
g
(
x
)
d
x
?
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What is the simplified form of the polynomial expression shown?
\newline
−
3
x
2
(
x
−
y
2
)
−
(
y
3
−
5
)
−
3
y
2
(
x
2
−
4
y
)
+
4
x
3
-3x^{2}(x-y^{2})-(y^{3}-5)-3y^{2}(x^{2}-4y)+4x^{3}
−
3
x
2
(
x
−
y
2
)
−
(
y
3
−
5
)
−
3
y
2
(
x
2
−
4
y
)
+
4
x
3
\newline
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∫
4
x
cos
(
2
−
3
x
)
d
x
\int 4 x \cos (2-3 x) d x
∫
4
x
cos
(
2
−
3
x
)
d
x
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(b) Use integration by parts to evaluate
∫
0
π
2
x
⋅
cos
x
d
x
\int_0^{\frac{\pi}{2}} x \cdot \cos x \, dx
∫
0
2
π
x
⋅
cos
x
d
x
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∫
sin
2
x
sin
x
d
x
\int \frac{\sin 2 x}{\sin x} d x
∫
s
i
n
x
s
i
n
2
x
d
x
=
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∫
x
3
−
2
x
x
d
x
\int \frac{x^{3}-2 \sqrt{x}}{x} d x
∫
x
x
3
−
2
x
d
x
=
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Simplify:
∫
x
x
d
x
\int \frac{x}{x}dx
∫
x
x
d
x
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∫
sec
t
(
sec
t
+
tan
t
)
d
t
\int \sec t(\sec t+\tan t) d t
∫
sec
t
(
sec
t
+
tan
t
)
d
t
=
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∫
x
(
x
−
1
)
5
d
x
\int x(x-1)^{5} d x
∫
x
(
x
−
1
)
5
d
x
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∫
x
x
+
1
\int \frac{\sqrt{x}}{x+1}
∫
x
+
1
x
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Evaluate the integral:
∫
2
x
x
2
−
36
d
x
\int \frac{2}{x \sqrt{x^{2}-36}} d x
∫
x
x
2
−
36
2
d
x
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Given the function
f
(
x
)
=
∣
x
−
3
∣
f(x)=|x-3|
f
(
x
)
=
∣
x
−
3∣
, what is the value of
∫
2
5
f
(
x
)
d
x
\int_{2}^{5} f(x) d x
∫
2
5
f
(
x
)
d
x
?\
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∫
cos
x
x
d
x
\int \frac{\cos \sqrt{x}}{\sqrt{x}} d x
∫
x
c
o
s
x
d
x
=
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∫
3
∞
1
x
x
2
−
9
d
x
\int_{3}^{\infty}\frac{1}{x\sqrt{x^{2}-9}}dx
∫
3
∞
x
x
2
−
9
1
d
x
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∫
2
4
2
x
x
2
−
4
d
x
\int_{2}^{4}\frac{2}{x\sqrt{x^{2}-4}}dx
∫
2
4
x
x
2
−
4
2
d
x
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Use the graph of the integrand to evaluate the integral.
\newline
∫
−
1
1
(
1
+
1
−
x
2
)
d
x
\int_{-1}^{1}(1+\sqrt{1-x^{2}})\,dx
∫
−
1
1
(
1
+
1
−
x
2
)
d
x
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Solve:
∫
ln
(
x
)
+
5
x
d
x
\int \frac{\ln(x) + 5}{x} \, dx
∫
x
l
n
(
x
)
+
5
d
x
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Find integral:
∫
x
(
x
)
d
x
\int x^{(x)}\,dx
∫
x
(
x
)
d
x
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The volume of the solid obtained by rotating the region enclosed by
\newline
y
=
x
2
,
x
=
y
2
y=x^{2}, \quad x=y^{2}
y
=
x
2
,
x
=
y
2
\newline
about the line
x
=
−
4
x=-4
x
=
−
4
can be computed using the method of disks or washers via an integral
\newline
V
=
∫
a
b
V=\int_{a}^{b}
V
=
∫
a
b
\newline
\newline
with limits of integration
a
=
a=
a
=
and
b
=
b=
b
=
\newline
The volume is
V
=
V=
V
=
cubic units.
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∫
cos
x
d
x
sin
2
x
+
4
sin
x
−
1
\int \frac{\cos x d x}{\sqrt{\sin ^{2} x+4 \sin x-1}}
∫
s
i
n
2
x
+
4
s
i
n
x
−
1
c
o
s
x
d
x
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Integration techniques: Evaluate
∫
(
x
2
+
3
x
−
2
)
d
x
\int (x^2 + 3x - 2) \, dx
∫
(
x
2
+
3
x
−
2
)
d
x
using appropriate integration methods.
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∫
x
d
x
+
∫
2
d
x
\int x d x+\int 2 d x
∫
x
d
x
+
∫
2
d
x
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Solve the equation. Check your solution
\newline
19
=
2
−
(
z
+
5
)
19=2-(z+5)
19
=
2
−
(
z
+
5
)
\newline
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∫
x
9
−
4
x
2
d
x
\int \frac{x}{\sqrt{9-4 x^{2}}} d x
∫
9
−
4
x
2
x
d
x
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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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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
∫
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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∫
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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∫
(
x
−
3
)
d
x
x
+
3
\int \frac{(x-3)\,dx}{x+3}
∫
x
+
3
(
x
−
3
)
d
x
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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
)
=
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
)
=
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
)
=
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 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
)
=
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 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
)
=
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
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
∫
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
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
∫
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
∫
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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