Statement-1: Let I n = ∫ 0 1 ( 1 − x 5 ) n dx . Then I 11 I 10 = 56 55 . Statement-2: If u ( x ) and v ( x ) are differentiable function, then ∫ u d v = uv − ∫ v d u + C , where C is const

Question

Statement-1: Let I n ​ = ∫ 0 1 ​ ( 1 − x 5 ) n dx . Then I 11 ​ I 10 ​ ​ = 56 55 ​ . Statement-2: If u ( x ) and v ( x ) are differentiable function, then ∫ u d v = uv − ∫ v d u + C , where C is constant of integration .

TestHub · Accepted Answer

I 11 ​ = ∫ 0 1 ​ I ( 1 − x 5 ) ​ ​ 11 ⋅ II 1 ​ ​ dx = ( 1 − x 5 ) 11 ⋅ x ] 0 1 ​ + 11 ∫ 0 1 ​ ( 1 − x 5 ) 10 5 x 4 ⋅ xdx I 11 ​ = 0 − 55 ∫ 0 1 ​ ( 1 − x 5 ) 10 ( 1 − x 5 − 1 ) d x = − 55 ∫ 0 1 ​ ( 1 − x 5 ) 11 d x + 55 I 10 ​ 56 I 11 ​ = 55 I 10 ​ ⇒ I 11 ​ I 10 ​ ​ = 55 56 ​

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