If int_{0}^{1} x^{m} (1 - x)^{n} dx = k int_{ | vedclass

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(D) We are given the integral $I = \int_{0}^{1} x^{m} (1 - x)^{n} dx$. Using the property of definite integrals,$\int_{0}^{a} f(x) dx = \int_{0}^{a} f

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$1$

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(D) We are given the integral $I = \int_{0}^{1} x^{m} (1 - x)^{n} dx$. Using the property of definite integrals,$\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a - x) dx$,we substitute $x$ with $(1 - x)$: $I = \int_{0}^{1} (1 - x)^{m} (1 - (1 - x))^{n} dx$ $I = \int_{0}^{1} (1 - x)^{m} (x)^{n} dx$ $I = \int_{0}^{1} x^{n} (1 - x)^{m} dx$. Comparing this with the given equation $\int_{0}^{1} x^{m} (1 - x)^{n} dx = k \int_{0}^{1} x^{n} (1 - x)^{m} dx$,we see that $I = k \cdot I$. Since the integral is non-zero,we conclude that $k = 1$.

If $\int_{0}^{1} x^{m} (1 - x)^{n} dx = k \int_{0}^{1} x^{n} (1 - x)^{m} dx$,then the value of $k$ equals

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