JEE MAIN - Mathematics (2006 - No. 2)
The value of $$\int\limits_1^a {\left[ x \right]} f'\left( x \right)dx,a > 1$$ where $${\left[ x \right]}$$ denotes the greatest integer not exceeding $$x$$ is
$$af\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .............f\left( {\left[ a \right]} \right)} \right\}$$
$$\left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + ...........f\left( {\left[ a \right]} \right)} \right\}$$
$$\left[ a \right]f\left( {\left[ a \right]} \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + ...........f\left( a \right)} \right\}$$
$$af\left( {\left[ a \right]} \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right) + .............f\left( a \right)} \right\}$$
Explanation
Let $$a = k + h$$ where $$k$$ is an integer such that
$$\left[ a \right] = k$$ and $$0 \le h < 1$$
$$\therefore$$ $$\int\limits_1^a {\left[ x \right]f'\left( x \right)dx = \int\limits_1^2 {1f'\left( x \right)} } \,dx$$
$$\,\,\,\,\,\,\,\,\,\,\,\, + \int\limits_2^3 {2f'\left( x \right)dx + } ....$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\int\limits_{k - 1}^k {\left( {k - 1} \right)dx + \int\limits_k^{k + h} {kf'\left( x \right)} } dx$$
$$\left\{ {f\left( 2 \right) - f\left( 1 \right)} \right\} + 2\left\{ {f\left( 3 \right) - f\left( 2 \right)} \right\}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ + 3\left\{ {f\left( 4 \right) - f\left( 3 \right)} \right\} + \,........\,$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ + \left( {k - 1} \right)\left\{ {f\left( k \right) - f\left( {k - 1} \right)} \right\}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ + k\left\{ {f\left( {k + h} \right) - f\left( k \right)} \right\}$$
$$ = - f\left( 1 \right) - f\left( 2 \right) - f\left( 3 \right).......$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ - f\left( k \right) + kf\left( {k + h} \right)$$
$$ = \left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right)} \right.$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\left. { + f\left( 3 \right) + ........f\left( {\left[ a \right]} \right)} \right\}$$
$$\left[ a \right] = k$$ and $$0 \le h < 1$$
$$\therefore$$ $$\int\limits_1^a {\left[ x \right]f'\left( x \right)dx = \int\limits_1^2 {1f'\left( x \right)} } \,dx$$
$$\,\,\,\,\,\,\,\,\,\,\,\, + \int\limits_2^3 {2f'\left( x \right)dx + } ....$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\int\limits_{k - 1}^k {\left( {k - 1} \right)dx + \int\limits_k^{k + h} {kf'\left( x \right)} } dx$$
$$\left\{ {f\left( 2 \right) - f\left( 1 \right)} \right\} + 2\left\{ {f\left( 3 \right) - f\left( 2 \right)} \right\}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ + 3\left\{ {f\left( 4 \right) - f\left( 3 \right)} \right\} + \,........\,$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ + \left( {k - 1} \right)\left\{ {f\left( k \right) - f\left( {k - 1} \right)} \right\}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ + k\left\{ {f\left( {k + h} \right) - f\left( k \right)} \right\}$$
$$ = - f\left( 1 \right) - f\left( 2 \right) - f\left( 3 \right).......$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ - f\left( k \right) + kf\left( {k + h} \right)$$
$$ = \left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right)} \right.$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\left. { + f\left( 3 \right) + ........f\left( {\left[ a \right]} \right)} \right\}$$
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