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Calculus
Gaussian integral
Evaluate the integral :
∫
−
∞
∞
e
x
(
1
+
e
x
)
d
x
\int_{-\infty}^{\infty} \frac{e^{x}}{\left(1+e^{x}\right)} d x
∫
−
∞
∞
(
1
+
e
x
)
e
x
d
x
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∫
x
2
(
6
−
x
3
)
5
d
x
\int x^{2}\left(6-x^{3}\right)^{5} d x
∫
x
2
(
6
−
x
3
)
5
d
x
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both positive, write the followi
\newline
y
3
45
x
7
y
5
y^{3} \sqrt{45 x^{7} y^{5}}
y
3
45
x
7
y
5
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∫
2
π
4
π
sin
2
x
d
x
\int_{2 \pi}^{4 \pi} \sin ^{2} x d x
∫
2
π
4
π
sin
2
x
d
x
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(
2.7
×
1
0
4
)
−
(
4
×
1
0
−
3
)
\sqrt{\left(2.7 \times 10^{4}\right)-\left(4 \times 10^{-3}\right)}
(
2.7
×
1
0
4
)
−
(
4
×
1
0
−
3
)
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Simplify the expression
\newline
(
log
n
p
)
4
+
(
log
p
n
)
4
+
2
+
2
−
log
n
p
−
log
p
n
\sqrt{\sqrt{(\log_{n}p)^{4}+(\log_{p}n)^{4}+2}+2}-\log_{n}p-\log_{p}n
(
lo
g
n
p
)
4
+
(
lo
g
p
n
)
4
+
2
+
2
−
lo
g
n
p
−
lo
g
p
n
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Simplify :
\newline
2
9
×
3
5
×
5
4
2
7
×
3
4
×
5
3
\frac{2^{9}\times3^{5}\times5^{4}}{2^{7}\times3^{4}\times5^{3}}
2
7
×
3
4
×
5
3
2
9
×
3
5
×
5
4
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∫
3
t
−
4
(
2
+
4
t
−
3
)
−
3
\int 3t^{-4}(2+4t^{-3})^{-3}
∫
3
t
−
4
(
2
+
4
t
−
3
)
−
3
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Integrate.
\newline
∫
3
t
−
4
(
2
+
4
t
−
3
)
−
3
\int 3 t^{-4}\left(2+4 t^{-3}\right)^{-3}
∫
3
t
−
4
(
2
+
4
t
−
3
)
−
3
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∫
3
t
−
4
(
2
+
4
t
−
3
)
−
3
\int 3 t^{-4}\left(2+4 t^{-3}\right)^{-3}
∫
3
t
−
4
(
2
+
4
t
−
3
)
−
3
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1
1
1
.
∫
e
−
3
x
d
x
\int e^{-3 x} d x
∫
e
−
3
x
d
x
\newline
4
4
4
.
∫
sin
3
x
d
x
\int \sin 3 x d x
∫
sin
3
x
d
x
\newline
2
2
2
.
∫
2
−
3
x
d
x
\int 2^{-3 x} d x
∫
2
−
3
x
d
x
\newline
5
5
5
.
∫
cos
(
2
x
−
3
)
d
x
\int \cos (2 x-3) d x
∫
cos
(
2
x
−
3
)
d
x
\newline
3
3
3
.
∫
4
2
x
−
3
d
x
\int \frac{4}{2 x-3} d x
∫
2
x
−
3
4
d
x
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∫
(
x
2
+
2
x
+
100
)
e
−
3
x
d
x
\int\left(x^{2}+2 x+100\right) e^{-3 x} d x
∫
(
x
2
+
2
x
+
100
)
e
−
3
x
d
x
=
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∫
a
x
e
x
d
x
\int a^{x} e^{x} d x
∫
a
x
e
x
d
x
.
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∫
(
4
e
3
x
+
1
)
d
x
\int\left(4 e^{3 x}+1\right) d x
∫
(
4
e
3
x
+
1
)
d
x
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∫
tan
2
(
x
)
+
1
+
tan
(
x
)
sec
(
x
)
d
x
\int \tan^{2}(x) + 1 + \tan(x) \sec(x) \, dx
∫
tan
2
(
x
)
+
1
+
tan
(
x
)
sec
(
x
)
d
x
=
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∫
x
3
e
2
x
d
x
\int x^{3} e^{2 x} d x
∫
x
3
e
2
x
d
x
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8
c
20
32
c
4
\frac{\sqrt{8 c^{20}}}{\sqrt{32 c^{4}}}
32
c
4
8
c
20
\newline
Which of the following is equivalent to the given expression for all
c
≠
0
c \neq 0
c
=
0
?
\newline
Choose
1
1
1
answer:
\newline
(A)
1
2
c
4
\frac{1}{2} c^{4}
2
1
c
4
\newline
(B)
1
2
c
8
\frac{1}{2} c^{8}
2
1
c
8
\newline
(C)
1
4
c
16
\frac{1}{4} c^{16}
4
1
c
16
\newline
D
1
4
c
2
c
\frac{1}{4} c^{2} \sqrt{c}
4
1
c
2
c
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∫
cos
(
x
2
)
e
2
x
d
x
\int \cos(x^2)e^{2x}\,dx
∫
cos
(
x
2
)
e
2
x
d
x
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Simplify:
\newline
(
i
)
(
2
5
)
2
×
7
3
8
3
×
7
(i) \frac{(2^{5})^{2}\times 7^{3}}{8^{3}\times 7}
(
i
)
8
3
×
7
(
2
5
)
2
×
7
3
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∫
cot
3
x
csc
5
x
d
x
\int \cot^{3}x \csc^{5}x \, dx
∫
cot
3
x
csc
5
x
d
x
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∫
cos
2
(
x
)
csc
2
(
2
x
)
d
x
\int \cos^2(x) \csc^2(2x) \, dx
∫
cos
2
(
x
)
csc
2
(
2
x
)
d
x
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