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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 : 212 of 271
Marks: +1, -0
∫−55∣x+2∣ dx\int\limits_{-5}^{5} |x+2| \, dx
Solution:  
Let I=∫−55∣x+2∣ dxI = \int\limits_{-5}^{5} |x+2| \, dx
Define ∣x+2∣={−(x+2),if x+2<0 or x<−2x+2,if x+2≥0 or x≥−2|x+2| = \begin{cases} -(x+2), & \text{if } x+2 < 0 \text{ or } x < -2 \\ x+2, & \text{if } x+2 \ge 0 \text{ or } x \ge -2 \end{cases}
∴I=−∫−5−2(x+2) dx+∫−25(x+2) dx\therefore I = -\int\limits_{-5}^{-2} (x+2) \, dx + \int\limits_{-2}^{5} (x+2) \, dx
=−[(x+2)22]−5−2+[(x+2)22]−25= -\left[ \frac{(x+2)^2}{2} \right]_{-5}^{-2} + \left[ \frac{(x+2)^2}{2} \right]_{-2}^{5}
−[(−2+2)22−(−5+2)22]+[(5+2)22−(−2+2)22]-\left[ \frac{(-2+2)^2}{2} - \frac{(-5+2)^2}{2} \right] + \left[ \frac{(5+2)^2}{2} - \frac{(-2+2)^2}{2} \right]
=29= 29
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