A) \[af(a)-\{f(1)+f(2)+...f([a])\}\]
B) \[[a]f(a)-\{f(1)+f(2)+...f([a])\}\]
C) \[[a]f(a)-\{f(1)+f(2)+...f(a)\}\]
D) \[af(a)-\{f(1)+f(2)+...f(a)\}\]
Correct Answer: B
Solution :
[b]| We have, |
| \[I=\int\limits_{1}^{a}{[x]f'(x)\,dx}\] |
| \[I=\int\limits_{1}^{2}{f'(x)dx}+2\int\limits_{2}^{3}{f'(x)dx+3\int\limits_{3}^{4}{f'(x)dx}}\]\[+...+[a]\int\limits_{{}}^{a}{[a]f'(x)dx}\] |
| \[I=[f\,(x)_{1}^{2}+2\,[f\,(x)_{2}^{3}+...+[a]f\,(x)_{a-1}^{a}\] |
| \[I=f\,(2)-f\,(1)+2\,(f)\,3-2f\,(2)+3f\,(3)\] |
| \[-3f\,(2)+...[a][f(a)-f[a]]\] |
| \[I=[a]f\,(a)-\{f\,(1)+f\,(2)+...f\,([a])\}\] |
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