a-b-c-d-0-1-sin-a-pix-cos-b-pix-x-c-1-x-d-dx-1-1-1-1-

Question Number 613 by 123456 last updated on 11/Feb/15

Ξ (a, b, c, d) = \underset{0}{\int^{1}} \frac{\sin^{a} (π x) \cos^{b} (π x)}{x^{c} {(1 - x)}^{d}} d x

Ξ (1, 1, 1, 1) = ?

Commented by prakash jain last updated on 10/Feb/15

Initial simplificaion

Ξ (1, 1, 1, 1) = \int_{0}^{1} \frac{\sin π x \cos π x}{x (1 - x)}

Ξ (a, b, 0, 0) = \int_{0}^{1} \sin^{a} (π x) \cos^{b} (π x) d x

Ξ (0, 0, 1 - c, 1 - d) = \int_{0}^{1} \frac{x^{c} {(1 - x)}^{d}}{x (1 - x)} \sin π x \cos π x d x

Commented by prakash jain last updated on 11/Feb/15

\int_{0}^{1} \frac{\sin (π x) \cos (π x)}{x (1 - x)} = \int_{0}^{1} \frac{\sin 2 π x}{x (1 - x)}

x = 0.5 - u \Rightarrow d x = - d u, x = 0, u = 0.5, x = 1, u = - 0.5

= - \int_{. 5}^{0.5} \frac{\sin (π - 2 π u)}{(0.5 - u) (0.5 + u)}

= \int_{- . 5}^{0.5} \frac{\sin 2 π u}{{(0.5)}^{2} - u^{2}} d x = 0 (∵ odd function of u)