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Evaluate: ∫_0^5(|x - 1| + |x - 2| + |x - 5|)dx - Mathematics

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प्रश्न

Evaluate:

`int_0^5(|x - 1| + |x - 2| + |x - 5|)dx`

मूल्यांकन
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उत्तर

`int_0^5(|x - 1| + |x - 2| + |x - 5|) dx`

I = `int_0^5[-(x - 1) - (x - 2) - (x - 5)] dx + int_1^2[(x - 1) - (x - 2) - (x - 5)] dx + int_2^5[(x - 1) + (x - 2) - (x - 5)] dx`

I = `int_0^1(-3x + 8)dx + int_1^2(-x + 6)dx + int_2^5 (x + 2) dx`

I = `((-3x^2)/2 + 8x)_0^1 + ((-x^2)/2 + 6x)_1^2 + ((x^2)/2 + 2x)_2^5`

= `((-3)/2 + 8 - 0) + ((-4)/2 + 12 + 1/2 - 6) + (25/2 + 10 - 4/2 - 4)`

= `13/2 + 9/2 + 33/2`

= `55/2` sq. units

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