Evaluate: `5050(int0 1(1-x^(50))^(100)dx)/(int0 1(1-x^(50))^(101)dx)`

Options

A. 5049
B. 5051
C. 5050
D. none of these

Answer

B

Explanation

Let `I_(1)=underset(0)overset(1)int (1-x^(50))^(100) x dx and , I_(2) =underset(0)overset(1)int(1-x^(50))^(101)dx`
Now,
`I_(2)=underset(0)overset(1)int (1-x^(50))^(101)dx=underset(0)overset(1)int underset(I) ((1-x^(50))^(100))underset(II)(1)dx`
`rArr I_(2)=[(1-x^(50))^(100)x]_(0)^(1)+5050+underset(0)overset(1)int(1-x^(50))^(100) x^(50) dx`
`rArr I_(2)=-5050underset(0)overset(1)int(1-x^(50))^(100){(1-x^(50))-1} dx`
`rArr I_(2)=-5050underset(0)overset(1)int(1-x^(50))^(101) dx+5050 underset(0)overset(1)int (1-x^(50))^(100)dx`
`rArr I_(2) -5050I_(2)+5050 I_(1)`
`rArr (5050 I_(1))/(I_(2)) =5051`