Problem 59 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int_{0}^{\pi}\left[\tan ^{-1} x\right] d x\) is equal to (where \([-]\) denotes greatest integer function) (A) \(\pi\) (B) \(\tan 1\) (C) \(\pi+\tan 1\) (D) \(\pi-\tan 1\)

Step-by-Step Solution

Answer

0

Step 1: Define the Problem

We need to evaluate the integral \( \int_{0}^{\pi}\left[\tan^{-1} x\right] dx \) where \([ \cdot ]\) represents the greatest integer function. This means we need to integrate \(\tan^{-1} x\) while applying the floor function to it.

Step 2: Identify the Range of Values

The greatest integer function \([\tan^{-1} x]\) will be evaluated by considering the integer part of \(\tan^{-1} x\) within the limits \(0\) and \(\pi\). The values we are interested in are obtained from \(\tan^{-1} x\) going from 0 at \(x=0\) to \(\frac{\pi}{2}\) at \(x=\infty\). At \(x=\pi\), \(\tan^{-1}\pi < 1\), resulting in \([\tan^{-1} x]=0\) over the entire interval \([0, \pi]\).

Step 3: Evaluate the Integral

Since \([\tan^{-1} x]=0\) for all \(x\) in \(0, \pi\), we integrate \(0\) over the interval from \(0\) to \(\pi\). This simplifies to \(\int_{0}^{\pi} 0 \cdot dx = 0\).

Step 4: Conclude the Solution

As the greatest integer function results in a value of 0 throughout the integration interval, the evaluated integral is zero: \(\int_{0}^{\pi} 0 \cdot dx = 0\). Thus, the value of the integral is 0.

Key Concepts

Greatest Integer Function

The Greatest Integer Function, also known as the floor function, is a mathematical function that maps a real number to the largest integer less than or equal to it. In simpler terms, it "rounds down" a number to the nearest whole number. This function is denoted by the square bracket notation \[ \left[ \cdot \right] \]. For example:

• \( \left[ 2.3 \right] = 2 \)
• \( \left[ -1.7 \right] = -2 \)
• \( \left[ 3 \right] = 3 \)

This creates step-like graphs where each segment is a horizontal line.

In integral calculus, using the greatest integer function means recognizing a discrete set of values for a continuous function. The function evaluates \[ \tan^{-1} x \] in the given exercise and maps each computed arctangent value to its closest lower integer. Because the function remains constant between integer points, it can simplify complex problems by reducing the range of possible outcomes, as seen in the provided exercise.

Inverse Trigonometric Function

Inverse trigonometric functions are used to find angles when the values of trigonometric functions are known. In this case, \[ \tan^{-1}x \], also expressed as "arctan" or "atan," is the inverse function of the tangent.

The function \[ \tan^{-1}x \] returns the angle in radians whose tangent is \( x \). It helps in converting a tangent value back to the angle itself, ranging from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) (not including these endpoints). This function illustrates a curve gently increasing through the origin.

In the exercise provided, we're concerned with this function as \( x \) ranges from 0 to \( \pi \). We observe that the output of \[ \tan^{-1}x \] is within the interval \( [0, 1) \), specifically resulting in the integer 0 when the greatest integer function is applied. This feature of the range simplifies the integral enormously.

Calculating Integrals

Calculating integrals involves summing up continuous measurements to find areas, accumulated quantities, or changes over an interval. In definite integrals, the task is computing the integral between specified bounds: in this case, from 0 to \( \pi \).

Here, using the greatest integer function altered the function being integrated into a constant, simplifying the original problem significantly. After recognizing the output of \[ \left[ \tan^{-1} x \right] \] as zero within the integration limits, the integration becomes trivial: integrating zero over any interval results in zero.

The outcome of a definite integral \[ \int_{a}^{b} f(x) \, dx \] is literally the "net area" under the curve from \( x = a \) to \( x = b \). That's why, if the function is zero, the area is also zero, confirming the evaluated result matches the exercise's conclusion.