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NCERT Class XII Mathematics Chapter - - Solutions

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Exercise - 7.11
By using the properties of definite integrals, evaluate the integrals in Exercises 208 to 225.
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Question : 225 of 271
Marks: +1, -0
∫28∣x−5∣dx\int\limits_{2}^{8} |x-5| dx
Solution:  
Let I=∫28∣x−5∣dxI=\int\limits_{2}^{8} |x-5| dx
Define ∣x−5∣={−(x−5)if x−5<0 or x<5(x−5)if x−5≥0 or x≥5|x-5| = \begin{cases} -(x-5) & \text{if } x-5 < 0 \text{ or } x<5 \\ (x-5) & \text{if } x-5 \ge 0 \text{ or } x\ge 5 \end{cases}
=−∫25(x−5)dx+∫58(x−5)dx=-\int\limits_{2}^{5} (x-5) dx + \int\limits_{5}^{8} (x-5) dx
=−[x22−5x]25+[x22−5x]58=-\left[ \frac{x^2}{2} - 5x \right]_{2}^{5} + \left[ \frac{x^2}{2} - 5x \right]_{5}^{8}
=−12(25−4)+5(5−2)+12(64−25)−5(8−5)=-\frac{1}{2}(25-4) + 5(5-2) + \frac{1}{2}(64-25) - 5(8-5)
=9=9
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