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Plz help me out in this definite integration PlzzzzZzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzZzzzzzzzzzzzzzzzzzzZzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz

Plz help me out in this definite integration
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Grade:11

1 Answers

Aditya Gupta
2080 Points
2 years ago
the answer will be n – 1.
here, we will use the formula ∫0 to n*T f(x)dx = n*∫0 to T f(x)dx where T is the period of the function and n is an integer.
now note that f(x)= {x} is a periodic function with period T= 1. so ∫0 to n {x}dx = n*∫0 to 1 {x}dx= n*∫0 to 1 xdx (as between 0 to 1 {x}= x)= n*x^2/2= n*1/2= n/2
now write ∫0 to n [x]dx= ∫0 to n xdx – ∫0 to n {x}dx= x^2/2 – n/2= n^2/2 – n/2
now divide the two (n^2/2 – n/2)/(n/2)
n – 1.
kindly approve :)

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