Q101 / 180 · KCET 2017 · Mathematics
\( \int_{-5}^{5}|x+2| d x \) is equal to
- Correct answerA\( 29 \)
- B(1)
- C\( 27 \)
- D\( 30 \)
Solution
Given that, \( \int_{-5}^{5}|x+2| d x \)
\[ \begin{array}{l} =\int_{-5}^{-2}|x+2| d x+\int_{-2}^{5}|x+2| d x \\ =-\int_{-5}^{-2}(x+2) d x+\int_{-2}^{5}(x+2) d x \\ =-\left[\frac{x^{2}}{2}+2 x\right]_{-5}^{-2}+\left[\frac{x^{2}}{2}+2 x\right]_{-2}^{5} \\ =-\left(\frac{4-25}{2}\right)-2(-2+5)+\frac{25-4}{2}+2(5+2) \\ =\frac{21}{2}+\frac{21}{2}-6 \pm 4=21+8=29 \end{array} \]
\[ \begin{array}{l} =\int_{-5}^{-2}|x+2| d x+\int_{-2}^{5}|x+2| d x \\ =-\int_{-5}^{-2}(x+2) d x+\int_{-2}^{5}(x+2) d x \\ =-\left[\frac{x^{2}}{2}+2 x\right]_{-5}^{-2}+\left[\frac{x^{2}}{2}+2 x\right]_{-2}^{5} \\ =-\left(\frac{4-25}{2}\right)-2(-2+5)+\frac{25-4}{2}+2(5+2) \\ =\frac{21}{2}+\frac{21}{2}-6 \pm 4=21+8=29 \end{array} \]
Hence, the correct answer is option: 'A'.