If ₀¹⁰ f(x) d x=5 , then _ k=1 ¹⁰ ₀^1 f(k-1+x) d x=

Step-by-step solution

Let I= ₀^1 f(k-1+x) d x Put k-1+x=t d x=d t Limits: aligned & x=0 t=k-1 & x=1 t=k aligned I= _ k-1 ^k f(t) d t It can also be written as, aligned & I= _ k-1 ^k f(x) d x & Now, _ k=1 ¹⁰ ₀^1 f(k-1+x) d x & = _ k=1 ¹⁰ _ k-1 ^k f(x) d x & = ₀^1 f(x) d x+ ₁^2 f(x) d x+ .+ ₉¹⁰ f(x) d x & = ₀¹⁰ f(x) d x=5 (from Eq. (i)) aligned