A) \[\int_{-100}^{100}{f({{x}^{2}})}\,dx\]
B) \[\int_{-100}^{100}{f(-{{x}^{2}})}\,dx\]
C) \[\int_{-100}^{100}{f\left( \frac{1}{x} \right)}\,dx\]
D) \[\int_{-100}^{100}{f(-x)}\,dx\]
E) \[\int_{-100}^{100}{[f(x)+f(-x)}]\,dx\]
Correct Answer: D
Solution :
\[\int_{-100}^{100}{f(x)dx}=2\int_{0}^{100}{f(x)}dx\] if \[f(-x)=f(x)\] ?.(i) \[=2\int_{0}^{100}{f(-x)}dx\] \[=\int_{-100}^{100}{f(-x)}dx\] Alternate \[=\int_{-100}^{100}{f(x)}dx=\int_{-100}^{100}{f(100-100-x)}dx\] \[=\int_{-100}^{100}{f(x)}dx\] [Using the property\[\int_{a}^{b}{f(x)}dx\] \[=\int_{a}^{b}{f(a+b-x)}dx]\]
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