Evaluate , and is independent of . Hence, show _ 0 ^ 1 x^k (1-x)^ n-k dx = 1 ^n C_k…

MATHEMATICS · Subjective · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Evaluate

\int_{0}^{1} (tx + 1 - x)^n dx

n \in N

and

t

is independent of

x

. Hence, show

\int_{0}^{1} x^k (1-x)^{n-k} dx = \frac{1}{{}^n C_k (n+1)}.