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Mathematics
Problem
The integral
∫
e
3
log
e
2
x
+
5
e
2
log
e
2
x
e
4
log
e
x
+
5
e
3
log
e
x
−
7
e
2
log
e
x
d
x
\int \frac{e^{3\log_e 2x} + 5e^{2\log_e 2x}}{e^{4\log_e x} + 5e^{3\log_e x} - 7e^{2\log_e x}} dx
∫
e
4
l
o
g
e
x
+
5
e
3
l
o
g
e
x
−
7
e
2
l
o
g
e
x
e
3
l
o
g
e
2
x
+
5
e
2
l
o
g
e
2
x
d
x
,
x
>
0
x > 0
x
>
0
, is equal to (where
c
c
c
is a constant of integration). (a)
log
e
∣
x
2
+
5
x
−
7
∣
+
c
\log_e |x^2 + 5x - 7| + c
lo
g
e
∣
x
2
+
5
x
−
7∣
+
c
(b)
4
log
e
∣
x
2
+
5
x
−
7
∣
+
c
4\log_e |x^2 + 5x - 7| + c
4
lo
g
e
∣
x
2
+
5
x
−
7∣
+
c
(c)
1
4
log
e
∣
x
2
+
5
x
−
7
∣
+
c
\frac{1}{4}\log_e |x^2 + 5x - 7| + c
4
1
lo
g
e
∣
x
2
+
5
x
−
7∣
+
c
(d)
log
e
∣
x
2
+
5
x
−
7
∣
+
c
\log_e \left| \sqrt{x^2 + 5x - 7} \right| + c
lo
g
e
x
2
+
5
x
−
7
+
c
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