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Mathematics
Problem
Let
K
n
=
∫
x
n
e
k
x
d
x
K_n=\displaystyle\int x^{n}e^{kx}\,dx
K
n
=
∫
x
n
e
k
x
d
x
with constant
k
≠
0
k\ne 0
k
=
0
and integer
n
≥
1
n\ge 1
n
≥
1
. Which of the following identities correctly relates
K
n
K_n
K
n
to
K
n
−
1
K_{n-1}
K
n
−
1
? a)
K
n
=
x
n
e
k
x
k
+
n
k
K
n
−
1
K_n=\dfrac{x^{n}e^{kx}}{k}+\dfrac{n}{k}\,K_{n-1}
K
n
=
k
x
n
e
k
x
+
k
n
K
n
−
1
b)
K
n
=
x
n
e
k
x
k
−
n
k
K
n
−
1
K_n=\dfrac{x^{n}e^{kx}}{k}-\dfrac{n}{k}\,K_{n-1}
K
n
=
k
x
n
e
k
x
−
k
n
K
n
−
1
c)
K
n
=
x
n
e
k
x
n
−
k
n
K
n
−
1
K_n=\dfrac{x^{n}e^{kx}}{n}-\dfrac{k}{n}\,K_{n-1}
K
n
=
n
x
n
e
k
x
−
n
k
K
n
−
1
d)
K
n
=
x
n
e
k
x
k
−
K
n
−
1
K_n=\dfrac{x^{n}e^{kx}}{k}-\,K_{n-1}
K
n
=
k
x
n
e
k
x
−
K
n
−
1
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