A) \[n.\int\limits_{0}^{n}{\mathbf{f}\left( \mathbf{x} \right)}.dx\]
B) \[\int\limits_{0}^{a}{\mathbf{f}\left( \mathbf{x} \right)}.dx\]
C) \[(n+1)\int\limits_{0}^{a}{\mathbf{f}\left( \mathbf{x} \right)}.dx\]
D) \[(n-1).\int\limits_{0}^{a}{\mathbf{f}\left( \mathbf{x} \right)}.dx\]
Correct Answer: D
Solution :
[d] \[\because I=\int\limits_{a}^{na}{f(x).dx}=\int\limits_{a}^{0}{f(x).dx}+\int\limits_{a}^{na}{f(x).dx}\] \[=-\int\limits_{a}^{a}{(x).dx}+\int\limits_{0}^{na}{f(x).dx}=-\int\limits_{0}^{a}{f(x).dx}+\int\limits_{0}^{a}{f(x).dx}\](By property of definite Integral) \[I=\left( n-1 \right).\int\limits_{a}^{a}{(x).dx}\] Hence option [d] is correct.
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