The value of ∫ limits01 cos ([2 x] π) cos (π x) d x is equal to [Note: [x] denotes the largest integer less than or equal to x.

Question

The value of ∫ limits01 cos ([2 x] π) cos (π x) d x is equal to [Note: [x] denotes the largest integer less than or equal to x. (A) 1 (B) (-2/π) (C) (2/π) (D) (1/π)

Tardigrade Admin · Accepted Answer

∫ limits01 cos ([2 x] π) cos (π x) d x=∫ limits01 / 2 cos π x d x+∫ limits1 / 21(- cos π x) d x=(1/π)+(1/π)=(2/π)

Q. The value of $0 \int 1 cos ([2 x] π) cos (π x) d x$ is equal to
[Note: $[x]$ denotes the largest integer less than or equal to $x$ .

1

$\frac{- 2}{π}$

$\frac{2}{π}$

$\frac{1}{π}$

$0 \int 1 cos ([2 x] π) cos (π x) d x = 0 \int 1/2 cos π x d x + 1/2 \int 1 (- cos π x) d x = \frac{1}{π} + \frac{1}{π} = \frac{2}{π}$

Q. The value of 0∫1​cos([2x]π)cos(πx)dx is equal to [Note: [x] denotes the largest integer less than or equal to x.

1

π−2​

π2​

π1​

0∫1​cos([2x]π)cos(πx)dx=0∫1/2​cosπxdx+1/2∫1​(−cosπx)dx=π1​+π1​=π2​

Q. The value of $0 \int 1 cos ([2 x] π) cos (π x) d x$ is equal to
[Note: $[x]$ denotes the largest integer less than or equal to $x$ .

$\frac{- 2}{π}$

$\frac{2}{π}$

$\frac{1}{π}$

$0 \int 1 cos ([2 x] π) cos (π x) d x = 0 \int 1/2 cos π x d x + 1/2 \int 1 (- cos π x) d x = \frac{1}{π} + \frac{1}{π} = \frac{2}{π}$