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Second Fundamental Theorem of Calculus

Which of the ...

Question

Which of the following is an odd function

sin x^{2}

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\frac{(a^{x} + 1)}{(a^{x} - 1)}

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x^{2} | x |

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none of these

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Solution

The correct option is B $\frac{(a^{x} + 1)}{(a^{x} - 1)}$
A. $f (x) = sin x^{2} \Rightarrow f (- x) = sin (- x) 2 = sin x^{2} = f (x) \Rightarrow$ even function
B. $f (x) = \frac{a^{x} + 1}{a^{x} - 1}$
$\Rightarrow f (- x) = \frac{a^{- x} + 1}{a^{- x} - 1} = \frac{1 + a^{x}}{1 - a^{x}} = - f (x) \Rightarrow$ odd function
C. $f (x) = x^{2} | x | \Rightarrow f (- x) = (- x)^{2} | - x | = x^{2} | x | = f (x) \Rightarrow$ even function
Hence only option 'B' is correct choice

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