Explanation
Step 1: Analyze the Greatest Integer Function [x3]
As x varies from −1 to 1, x3 also varies from −1 to 1. The value of [x3] changes at x3=0.
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For -1 \le x < 0:
-1 \le x^3 < 0 \implies [x^3] = -1
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For 0 \le x < 1:
0 \le x^3 < 1 \implies [x^3] = 0
Step 2: Split the Integral
We split the integral into two parts based on the intervals identified above:
I=∫−10x2e[x3]dx+∫01x2e[x3]dx
Substitute the values of [x3]:
I=∫−10x2e−1dx+∫01x2e0dx
Step 3: Evaluate the Integrals
Evaluating the first part:
∫−10x2e−1dx=e1∫−10x2dx
e1[3x3]−10=e1(0−(3−1))=3e1
Evaluating the second part:
Step 4: Final Summation
Add the results of the two parts:
Final Answer:
The value of the integral is 3ee+1.
