You may start by checking if the function is an odd or even function, such that:
Hence, the function is an odd function and you may use the following property, such that:
, if f(x) is odd
Hence, according to this property,
In his book Night , Elie Wiesel uses symbolism throughout to enhance the text. First of all, the title itself is symbolic. The word "ni...
%7D%7D%7D%3D%7B%7Bx%7D%7D%5E%7B%7B3%7D%7D%2B%7B%7Bx%7D%7D%5E%7B%7B4%7D%7D%5Ccdot%7B%5Ctan%7B%7Bx%7D%7D%7D "f(x) = x^3 + x^4*tan x")
is an odd or even function, such that:%7D%7D%7D%3D%7B%7B%5Cleft(-%7Bx%7D%5Cright)%7D%7D%5E%7B%7B3%7D%7D%2B%7B%7B%5Cleft(-%7Bx%7D%5Cright)%7D%7D%5E%7B%7B4%7D%7D%5Ccdot%7B%5Ctan%7B%7B%5Cleft(-%7Bx%7D%5Cright)%7D%7D%7D%3D-%7B%7Bx%7D%7D%5E%7B%7B3%7D%7D%2B%7B%7Bx%7D%7D%5E%7B%7B4%7D%7D%5Ccdot%7B%5Cleft(-%7B%5Ctan%7B%7Bx%7D%7D%7D%5Cright)%7D%3D-%7B%7Bx%7D%7D%5E%7B%7B3%7D%7D-%7B%7Bx%7D%7D%5E%7B%7B4%7D%7D%5Ccdot%7B%5Ctan%7B%7Bx%7D%7D%7D%3D-%7B%5Cleft(%7B%7Bx%7D%7D%5E%7B%7B3%7D%7D%2B%7B%7Bx%7D%7D%5E%7B%7B4%7D%7D%5Ccdot%7B%5Ctan%7B%7Bx%7D%7D%7D%5Cright)%7D%3D-%7Bf%7B%7B%5Cleft(%7Bx%7D%5Cright)%7D%7D%7D "f(-x) = (-x)^3 + (-x)^4*tan(-x) = -x^3 + x^4*(-tan x) = -x^3 - x^4*tan x = -(x^3 + x^4*tan x) = -f(x)")
is an odd function and you may use the following property, such that:%7D%7D%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D%3D%7B0%7D "int_(-a)^a f(x) dx = 0")
, if f(x) is odd%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D%3D%7B0%7D. "int_(-pi/4)^(pi/4) (x^3 + x^4*tan x)dx = 0.")
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