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By using the properties of definite int...

By using the properties of definite integrals, evaluate the integrals`int_-5^ 5|x+2|dx`

Text Solution

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Let` I=int_(-5)^(5)∣x+2∣dx`
It can be seen that (x+2)≤0 on [−5,−2] and (x+2)≥0 on [−2,5].
∴`I=int_(-5)^(-2)​−(x+2)dx+int_(-2)^(5)(x+2)dx`
=`-int_(-5)^(-2)​[x^2/2​+2x]+int_(-2)^(5)​[x^2/2​+2x]`
=`−[((−2)^2​)/2+2(−2)−(−5)^2/2​−2(−5)]+[((5)^2)/2+2(5)−(−2)^2/2​−2(−2)]`
=`−[2−4−25/2​+10]+[25/2​+10−2+4]`
=`−2+4+25/2​−10+25/2​+10−2+4=29`
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