Animated Solution for Mathematics - Definite Integration: The value of ${\int }_1^a[x]f^{\prime }(x)dx,a>1$ where $[x]$ denotes the greatest integer not exceeding $x$ is

The Greatest Integer Function, denoted as $[x]$, behaves like a staircase. It remains constant at an integer $k$ for all $x$ in the interval $[k,k+1)$.

When evaluating the integral ${\int }_1^a[x]f^{\prime }(x)dx$, the discontinuity of $[x]$ prevents direct integration. We must respect its piecewise nature to solve the problem effectively.

Let $n=[a]$ be the largest integer less than or equal to $a$. We split the integral from $1$ to $a$ into discrete, manageable intervals: $[1,2),[2,3),\dots ,[n,a]$.

This transforms the integral into a sum of smaller, friendly components:

In each sub-interval, $[x]$ is a constant. We can pull these constants out of the integral signs. Applying the Fundamental Theorem of Calculus, where the integral of $f^{\prime }(x)$ is $f(x)$, we obtain:

This step successfully converts a calculus problem into an algebraic one.

Expanding the brackets, we get:

By grouping terms with the same function value $f(k)$, we observe a pattern. For any integer $k$ where $1<k\le n$, the term $f(k)$ appears with a coefficient of $(k-1)-k=-1$.

The expression simplifies significantly as the intermediate terms cancel out:

Factoring out the negative sign, we arrive at the final expression. Substituting $n=[a]$ back into the equation, we obtain the elegant result:

By breaking the problem down, we have uncovered the underlying structure. Whenever you encounter a step function, remember to partition the domain, integrate the pieces, and allow the telescoping sum to reveal the solution.