题目

判断下列函数的奇偶性-|||-(1) (x)=x(sin )^2x:-|||-(2) (x)=xsin (x)^2-|||-(3) (x)=sin (arctan x)-|||-(4) (x)=dfrac ({x)^3cos ((x)^3)}(1+{x)^2}

$\begin{aligned}&判断下列函数的奇偶性\\ &\left(1\right)f\left(x\right)=x\sin^{2}x:\\ &\left(2\right)f\left(x\right)=x\sin x^{2}\\ &\left(3\right)f\left(x\right)=sin(arctan x)\\ &(4)f(x)=\frac{x^3cos(x^3)}{1+x^2} \\&\end{aligned}$

题目解答

答案

$\begin{aligned}&（1）\\ & f(-x)=-x\sin^2(-x)\\ & =-x\sin^2x\\ & =-f(x) \\ &所以 f(x) = x\sin^2x 是奇函数。\\ &（2）\\ & f(-x)=-x\sin((-x)^2)\\ & =-x\sin(x^2)\\ & =-f(x) \\ &所以 f(x) = x\sin x^2 是奇函数。\\ &（3）\\ & f(-x)=\sin(\arctan(-x))\\ & =\sin(-\arctan(x))\\ & =-\sin(\arctan(x))\\ & =-f(x) \\ &所以 f(x) = \sin(\arctan x) 是奇函数。\\ &（4）\\ & f(-x)=\frac{(-x)^3\cos((-x)^3)}{1 + (-x)^2}\\ & =-\frac{x^3\cos(x^3)}{1 + x^2}\\ & =-f(x) \\ &所以 f(x) = \frac{x^3\cos(x^3)}{1 + x^2} 是奇函数。 \\&\end{aligned}$