Question:
Integration of greatest integer function of x square . Given upper limit = 2 ,
lower limit =0
Answer:
0∫2 [x2 ] dx = 0∫1 [x2 ] dx + 1∫√2 [x2 ] dx + 02∫√3 [x2 ] dx + √3∫2 [x2 ] dx
= 0∫1 0 dx + 1∫√2 1 dx + √2∫√3 2 dx + √3∫2 3 dx
= 0 + [x 1]√2 + 2[x √2]√3 + 3[x √3]2 +
= (√2 - 1) + 2(√3 - √2) + 3(2 - √3)
= √2 - 1 + 2√3 - 2√2 + 6 - 3√3
= 5 - √3 - √2
So, 0∫2 [x2 ] dx = 5 - √3 - √2
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