Solve-I-n-0-n-1-x-x-2-dx-

Question Number 6390 by FilupSmith last updated on 25/Jun/16

Solve :

I (n) = \int_{0}^{n} {(- 1)}^{⌊ x ⌋} x^{2} d x

Commented by prakash jain last updated on 25/Jun/16

I (n) = \int_{0}^{1} x^{2} d x - \int_{1}^{2} x^{2} d x + - \dots + {(- 1)}^{n - 1} \int_{n - 1}^{n} x^{2} d x

= \frac{1^{3}}{3} - (\frac{2^{3}}{3} - \frac{1^{3}}{3}) + (\frac{3^{3}}{3} - \frac{2^{3}}{3}) + - . . + {(- 1)}^{n - 1} (\frac{n^{3}}{3} - \frac{{(n - 1)}^{3}}{3})

Commented by FilupSmith last updated on 26/Jun/16

I love this approach! i didnt even think

of it this way!

Commented by malwaan last updated on 26/Jun/16

t h i s i s r i g h t f o r {(- 1)}^{n} n o t {(- 1)}^{∣ x ∣}

Commented by prakash jain last updated on 26/Jun/16

⌊ x ⌋

0 f o r 0 ⩽ x < 1

1 f o r 1 ⩽ x < 2

a n d s o o n