The integral ∫ 0 1 1 7 1 x d x , where · denotes the greatest integer function, is equal to

Step-by-step solution

I = ∫ 0 1 d x 7 1 x = ∫ 0 1 1 7 1 x d x Let 1 7 = k   ⇒ I = ∫ 0 1 k 1 x d x I = ∫ 0 1 k 1 x d x ⇒ I = ∫ 1 2 1 k 1 x d x + ∫ 1 3 1 2 k 1 x d x + ∫ 1 4 1 3 k 1 x d x + ⋯ ⇒     I = k 1 - 1 2 + k 2 1 2 - 1 3 + k 3 1 3 - 1 4 + ⋯ ⇒ I = k + k 2 2 + k 3 3 + ⋯ - 1 k k 2 2 + k 3 3 + ⋯ ⇒ I = - ln 1 - k - 1 k - ln 1 - k - k ⇒   I = - ln 6 7 - 7 - ln 6 7 - 1 7   ∵ k = 1 7 ∴ I = 1 + 6 ln 6 7