KCET · Maths · Functions
\( \int_{-5}^{5}|x+2| d x \) is equal to
- A \( 29 \)
- B (1)
- C \( 27 \)
- D \( 30 \)
Answer & Solution
Correct Answer
(A) \( 29 \)
Step-by-step Solution
Detailed explanation
Given that, \( \int_{-5}^{5}|x+2| d x \)
\[
\begin{array}{l}
=\int_{-5}^{-2}|x+2| d x+\int_{-2}^{5}|x+2| d x \\
=-\int_{-5}^{-2}(x+2) d x+\int_{-2}^{5}(x+2) d x \\
=-\left[\frac{x^{2}}{2}+2 x\right]_{-5}^{-2}+\left[\frac{x^{2}}{2}+2 x\right]_{-2}^{5} \\
=-\left(\frac{4-25}{2}\right)-2(-2+5)+\frac{25-4}{2}+2(5+2) \\
=\frac{21}{2}+\frac{21}{2}-6 \pm 4=21+8=29
\end{array}
\]
\[
\begin{array}{l}
=\int_{-5}^{-2}|x+2| d x+\int_{-2}^{5}|x+2| d x \\
=-\int_{-5}^{-2}(x+2) d x+\int_{-2}^{5}(x+2) d x \\
=-\left[\frac{x^{2}}{2}+2 x\right]_{-5}^{-2}+\left[\frac{x^{2}}{2}+2 x\right]_{-2}^{5} \\
=-\left(\frac{4-25}{2}\right)-2(-2+5)+\frac{25-4}{2}+2(5+2) \\
=\frac{21}{2}+\frac{21}{2}-6 \pm 4=21+8=29
\end{array}
\]
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