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Question
Evaluate: `int_(-1)^3 |x^3 - x|dx`
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Solution
Let I = `int_(-1)^2|x^3 - x|dx`
= `int_(-1)^2|x(x^2 - 1)|dx`
= `int_(-1)^2|x(x - 1)(x + 1)|dx`
Here, x3 – x = 0, when x = 0, 1, –1
| Value of x | Value of (x3 – x) |
| –1 < x < 0 | +ve |
| 0 < x < 1 | –ve |
| 1 < x < 2 | +ve |
∴ |x3 – x| = `{{:(x^3 - x, if -1 < x < 0 and 1 < x < 2),(-x^3 + x, if 0 < x < 1):}`
I = `int_(-1)^0(x^3 - x)dx + int_1^1(-x^3 + x)dx + int_1^2(x^3 - x)dx`
= `[x^4/4 - x^2/2]_-1^0 + [(-x^4)/4 + x^2/2]_0^1 + [x^4/4 - x^2/2]_1^2`
= `1/4 + 1/4 + 2 + 1/4`
= `2 + 3/4`
= `11/4`
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