CBSE Class 12 · 2013 · Definite IntegralsLong Answer

Evaluate:

\int_{2}^{5} [∣ x - 2∣ + ∣ x - 3∣ + ∣ x - 5∣] d x

Answer: $\frac{23}{2}$

Solution

Integrate each modulus over

[2, 5]

, removing the sign on each sub-interval.

\int_{2}^{5} ∣ x - 2∣ d x = \int_{2}^{5} (x - 2) d x = [\frac{( x - 2 ) ^{2}}{2}]_{2}^{5} = \frac{9}{2}

\int_{2}^{5} ∣ x - 3∣ d x = \int_{2}^{3} (3 - x) d x + \int_{3}^{5} (x - 3) d x = \frac{1}{2} + 2 = \frac{5}{2}

\int_{2}^{5} ∣ x - 5∣ d x = \int_{2}^{5} (5 - x) d x = [5 x - \frac{x ^{2}}{2}]_{2}^{5} = \frac{9}{2}

I = \frac{9}{2} + \frac{5}{2} + \frac{9}{2} = \frac{23}{2}

CBSE Class 12 Board · 2013 · Subjective