First slide

Evaluation of definite integrals

Question

$\int_{- 3 / 2}^{10} {2 x} dx$ where {.} denotes the fractional part of x, is equal to

Moderate

A

$\frac{1}{2}$
B

$\frac{13}{2}$
C

$\frac{23}{2}$
D

$\frac{23}{4}$

Solution

$f (x) = {2 x}$ is a periodic function with period $\frac{1}{2}$
Let
$\begin{array}{l} I = \int_{- 3 / 2}^{10} {2 x} dx = \int_{- 3 (1 / 2)}^{20 (1 / 2)} {2 x} dx \\ = 23 \int_{0}^{1 / 2} 2 xdx \\ [as {2 x} = 2 x - [2 x] and when \in [0, \frac{1}{2}), [2 x] = 0] \\ = 23 {[x^{2}]}_{0}^{1 / 2} = \frac{23}{4} \end{array}$

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