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Mathematics
Mathematics
Problem
Evaluate
∫
0
1
(
t
x
+
1
−
x
)
n
d
x
\int_{0}^{1} (tx + 1 - x)^n dx
∫
0
1
(
t
x
+
1
−
x
)
n
d
x
,
n
∈
N
n \in N
n
∈
N
and
t
t
t
is independent of
x
x
x
. Hence, show
∫
0
1
x
k
(
1
−
x
)
n
−
k
d
x
=
1
n
C
k
(
n
+
1
)
.
\int_{0}^{1} x^k (1-x)^{n-k} dx = \frac{1}{{}^n C_k (n+1)}.
∫
0
1
x
k
(
1
−
x
)
n
−
k
d
x
=
n
C
k
(
n
+
1
)
1
.
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