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By using the properties of definite int...

By using the properties of definite integrals, evaluate the integrals`int_0^4 |x-1|dx`

Text Solution

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`I=int_0 ^4∣x−1∣dx`
It can be seen that, (x−1)≤0 when 0≤x≤1 and (x−1)≥0 when 1≤x≤4
`I=int_0^1​∣x−1∣dx+int_1^4∣x−1∣dx,(∵∫ab​f(x)=∫ac​f(x)+∫cb​f(x))`
=`int_0^1​​−(x−1)dx+int_1^4​(x−1)dx`
=`[x−x^2/2​]_0^1​+[x^2/2​−x]_1^4`
=`1−1/2​+(4)^2/2​−4−1/2​+1`
=`1−1/2​+8−4−1/2​+1=5`
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