The value of _1^a [x]f'(x),dx, a1 where [x] denotes the greatest integer not exceeding x …

Question

The value of $\displaystyle\int_1^a [x]f'(x)\,dx$, $a>1$ where $[x]$ denotes the greatest integer not exceeding $x$ is:

Accepted Answer

Let $a = k + h$ where $k$ is an integer such that

$$\left[ a \right] = k$$ and $$0 \le h < 1$$

$$\therefore$$ $$\int\limits_1^a {\left[ x \right]f'\left( x \right)dx = \int\limits_1^2 {1f'\left( x \right)} } \,dx$$

$$\,\,\,\,\,\,\,\,\,\,\,\, + \int\limits_2^3 {2f'\left( x \right)dx + } ....$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\int\limits_{k - 1}^k {\left( {k - 1} \right)dx + \int\limits_k^{k + h} {kf'\left( x \right)} } dx$$

$$\left\{ {f\left( 2 \right) - f\left( 1 \right)} \right\} + 2\left\{ {f\left( 3 \right) - f\left( 2 \right)} \right\}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ + 3\left\{ {f\left( 4 \right) - f\left( 3 \right)} \right\} + \,........\,$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ + \left( {k - 1} \right)\left\{ {f\left( k \right) - f\left( {k - 1} \right)} \right\}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ + k\left\{ {f\left( {k + h} \right) - f\left( k \right)} \right\}$$

$$ =  - f\left( 1 \right) - f\left( 2 \right) - f\left( 3 \right).......$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ - f\left( k \right) + kf\left( {k + h} \right)$$

$$ = \left[ a \right]f\left( a \right) - \left\{ {f\left( 1 \right) + f\left( 2 \right)} \right.$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\left. { + f\left( 3 \right) + ........f\left( {\left[ a \right]} \right)} \right\}$$

Solutions · 1