∫$$12$$ $$x3$$−$$1dx$$ where [.] denotes the greatest integer function, is equal to

Question

∫$$12$$ $$x3$$−$$1dx$$ where [.] denotes the greatest integer function, is equal to

Infinity Learn · Accepted Answer

$$1$$≤x≤$$2$$⇒$$1$$≤$$x3$$≤$$8$$⇒$$0$$≤$$x3$$−$$1$$≤$$7I=$$∫$$12$$ $$x3$$−$$1dx=$$∫$$121/3$$ $$x3$$−$$1dx+$$∫$$21/331/3$$ $$x3$$−$$1dx+$$…$$+$$∫$$71/32$$ $$x3$$−$$1dx$$ now, x∈$$1$$,$$213$$, then $$x3$$∈[$$1$$,$$2)$$ or $$x3$$−$$1=0$$ and so ontherefore,$$I=$$∫$$121/3$$ $$0dx+$$∫$$21/331/3$$ $$1$$⋅$$dx+$$…∫$$71/32$$ $$6dx$$ $$=31/3$$−$$21/3+241/3$$−$$31/3+351/3$$−$$41/3+461/3$$−$$41/3+561/3$$−$$51/3+621/3$$−$$71/3=12$$−$$71/3+61/3+51/3+41/3+31/3+21/3$$

$\int_{1}^{2} [x^{3} - 1] dx$ where [.] denotes the greatest integer function, is equal to

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∫12 x3−1dx where [.] denotes the greatest integer function, is equal to

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$\int_{1}^{2} [x^{3} - 1] dx$ where [.] denotes the greatest integer function, is equal to