First slide

Evaluation of definite integrals

Question

The integral $I = \int_{- 3 / 2}^{3 / 2} ([x] + x^{3} + \log_{a} (x + \sqrt{x^{2} + 1})) d x$ is equal

Moderate

A

0
B

– 3/2
C

1
D

3/2

Solution

Let $f (x) = x^{3} + \log_{a} (x + \sqrt{x^{2} + 1})$
$\begin{array}{l} f (- x) = - x^{3} + \log_{a} (- x + \sqrt{x^{2} + 1}) \\ = - x^{3} + \log_{a} \frac{1}{x + \sqrt{x^{2} + 1}} \\ = - x^{3} + \log_{a} (x + \sqrt{x^{2} + 1}) = - f (x) \end{array}$
$∴ \int_{- 3 / 2}^{3 / 2} f (x) d x = 0$
Thus $I = \int_{- 3 / 2}^{3 / 2} [x] d x$
$\begin{array}{l} = \int_{- 3 / 2}^{- 1} (- 2) d x + \int_{- 1}^{0} - 1 d x + \int_{0}^{1} 0 d x + \int_{1}^{3 / 2} 1 d x \\ = (- 2) [- 1 + 3 / 2] + (- 1) [0 + 1] + 0 + 1 [3 / 2 - 1] \\ = - 1 - 1 + 1 / 2 = - 3 / 2 . \end{array}$

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