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JEE Main 2024 · Mathematics · Indefinite and Definite Integrals · medium

The value of ∫0^4(|x-1|+|x-3|)dx is

Numerical answer

The value of ∫04(∣x−1∣+∣x−3∣)dx\displaystyle\int_0^4\left(|x-1|+|x-3|\right)dx is
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Answer: 10

Split at the kinks. ∫04∣x−1∣ dx=∫01(1−x)dx+∫14(x−1)dx=12+92=5\int_0^4|x-1|\,dx=\int_0^1(1-x)dx+\int_1^4(x-1)dx=\tfrac12+\tfrac92=5. Similarly ∫04∣x−3∣ dx=∫03(3−x)dx+∫34(x−3)dx=92+12=5\int_0^4|x-3|\,dx=\int_0^3(3-x)dx+\int_3^4(x-3)dx=\tfrac92+\tfrac12=5. The sum is 1010.
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