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Math Problems
Calculus
Find indefinite integrals using the substitution
Evaluate the integral.
\newline
∫
−
x
e
−
2
x
d
x
\int-x e^{-2 x} d x
∫
−
x
e
−
2
x
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
3
x
cos
(
−
2
x
+
2
)
d
x
\int 3 x \cos (-2 x+2) d x
∫
3
x
cos
(
−
2
x
+
2
)
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
−
6
x
sin
(
−
x
)
d
x
\int-6 x \sin (-x) d x
∫
−
6
x
sin
(
−
x
)
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
−
4
x
sin
(
4
x
+
5
)
d
x
\int-4 x \sin (4 x+5) d x
∫
−
4
x
sin
(
4
x
+
5
)
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
5
x
e
x
+
6
d
x
\int 5 x e^{x+6} d x
∫
5
x
e
x
+
6
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
4
x
e
−
3
x
+
1
d
x
\int 4 x e^{-3 x+1} d x
∫
4
x
e
−
3
x
+
1
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
x
−
2
ln
(
−
x
)
d
x
\int x^{-2} \ln (-x) d x
∫
x
−
2
ln
(
−
x
)
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
−
x
4
−
3
x
d
x
\int-x 4^{-3 x} d x
∫
−
x
4
−
3
x
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
6
x
3
e
−
2
x
d
x
\int 6 x^{3} e^{-2 x} d x
∫
6
x
3
e
−
2
x
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
x
5
3
x
d
x
\int x 5^{3 x} d x
∫
x
5
3
x
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
−
2
x
e
2
x
d
x
\int-2 x e^{2 x} d x
∫
−
2
x
e
2
x
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
−
2
x
cos
(
−
2
x
)
d
x
\int-2 x \cos (-2 x) d x
∫
−
2
x
cos
(
−
2
x
)
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
−
x
cos
(
−
3
x
)
d
x
\int-x \cos (-3 x) d x
∫
−
x
cos
(
−
3
x
)
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
6
x
2
5
3
x
d
x
\int 6 x^{2} 5^{3 x} d x
∫
6
x
2
5
3
x
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
−
6
x
4
−
4
x
d
x
\int-6 x 4^{-4 x} d x
∫
−
6
x
4
−
4
x
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
6
x
2
2
3
x
d
x
\int 6 x^{2} 2^{3 x} d x
∫
6
x
2
2
3
x
d
x
\newline
Answer:
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Evaluate the integral.
\newline
∫
−
3
x
sin
(
−
2
x
)
d
x
\int-3 x \sin (-2 x) d x
∫
−
3
x
sin
(
−
2
x
)
d
x
\newline
Answer:
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y
=
x
4
d
y
d
x
=
\begin{array}{c}y=x^{4} \\ \frac{d y}{d x}=\end{array}
y
=
x
4
d
x
d
y
=
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Can this differential equation be solved using separation of variables?
\newline
d
y
d
x
=
3
x
2
y
−
8
y
\frac{d y}{d x}=\frac{3}{x^{2} y-8 y}
d
x
d
y
=
x
2
y
−
8
y
3
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
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Consider the curve given by the equation
x
y
2
−
x
3
y
=
8
x y^{2}-x^{3} y=8
x
y
2
−
x
3
y
=
8
. It can be shown that
d
y
d
x
=
3
x
2
y
−
y
2
2
x
y
−
x
3
\frac{d y}{d x}=\frac{3 x^{2} y-y^{2}}{2 x y-x^{3}}
d
x
d
y
=
2
x
y
−
x
3
3
x
2
y
−
y
2
.
\newline
Write the equation of the vertical line that is tangent to the curve.
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Consider the curve given by the equation
2
x
2
−
12
x
+
y
3
=
107
2 x^{2}-12 x+y^{3}=107
2
x
2
−
12
x
+
y
3
=
107
. It can be shown that
d
y
d
x
=
4
(
3
−
x
)
3
y
2
\frac{d y}{d x}=\frac{4(3-x)}{3 y^{2}}
d
x
d
y
=
3
y
2
4
(
3
−
x
)
.
\newline
Find the point on the curve where the line tangent to the curve is horizontal.
\newline
(
□
(\square
(
□
,
□
)
\square)
□
)
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Find
d
2
d
x
2
[
5
ln
(
x
5
)
]
\frac{d^{2}}{d x^{2}}\left[5 \ln \left(x^{5}\right)\right]
d
x
2
d
2
[
5
ln
(
x
5
)
]
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