The value of `int_a^b(x-a)^3(b-x)^4dxi s` `((b-a)^4)/(6^4)` (b) `((b-a)^8)/(280)` `((b-a)^7)/(7^3)` (d) none of these

Options

A. `((b-a)^(4))/(6^(4))`
B. `((b-a)^(8))/280`
C. `((b-a)^(7))/(7^(3))`
D. none of these

Answer

B

Explanation

Put `x=a cos^(2) theta+b sin^(2) theta` or `dx=2(b-a) sin theta cos theta d theta`. Then
`int_(a)^(b)(x-a)^(3)(b-x)^(4)dx`
`=2(b-a)int_(0)^(pi//2)(a cos^(2) theta+b sin^(2) theta-a)^(3)(b-a cos^(2)theta-bsin^(2) theta)^(4)sin theta cos theta d theta`
`=2(b-a)^(8)int_(0)^(pi//2) sin^(7) theta cos^(9) theta d theta`
`=2 (b-a)^(8) int_(0)^(pi//2) sin^(7) theta(1-sin^(2) theta)^(4) cos theta d theta`
`2(b-a)^(8) int_(0)^(1) x^(7) (1-x^(2))^(4)dx`
`=2(b-a)^(8) int_(0)^(1) x^(7)(1-4x^(2)+6x^(4)-4x^(6)+x^(8))dx`
`=2(b-a)^(8)[1/8-4/10+6/12-4/14+1/16]=((b-a)^(8))/280`