The value of ∫(-π/2 to π/2) (1/[x]+4)dx, where [.] denotes the greatest integer function, is

The greatest integer function, denoted by $[x]$ or floor$(x)$, gives the largest integer less than or equal to $x$. For example, $[3.7] = 3$, $[5] = 5$, $[-2.3] = -3$, and $[-1] = -1$. This function is also called the floor function. The graph of $y = [x]$ consists of horizontal line segments at integer heights, with jumps (discontinuities) at every integer value. Key properties include: $[x] \leq x < [x] + 1$, which means $x - 1 < [x] \leq x$. For any real number $x$ and integer $n$, we have $[x + n] = [x] + n$. The greatest integer function is neither continuous nor differentiable at integer points, but it is continuous from the right at every integer. When integrating functions involving $[x]$, we must break the integral into intervals where $[x]$ remains constant. Within each such interval, $[x]$ is simply a constant, making the integration straightforward. The function has a step-like behavior and is particularly useful in problems involving discrete quantities or when modeling situations with threshold effects. In JEE, problems often combine $[x]$ with other functions, requiring careful analysis of the domain and recognition of intervals where the floor function takes constant values. Understanding the periodicity and symmetry properties of $[x]$ can greatly simplify calculations in definite integrals.

Definite integration computes the signed area under a curve between two limits. The fundamental theorem of calculus states that $\int_a^b f(x)dx = F(b) – F(a)$, where $F'(x) = f(x)$. For piecewise functions or functions involving the greatest integer function, we break the integral into subintervals where the function has a simpler form. Properties of definite integrals include linearity: $\int_a^b [cf(x) + dg(x)]dx = c\int_a^b f(x)dx + d\int_a^b g(x)dx$, and additivity over intervals: $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$ for $a < c < b$. For symmetric limits $[-a, a]$, if $f(x)$ is an even function (meaning $f(-x) = f(x)$), then $\int_{-a}^a f(x)dx = 2\int_0^a f(x)dx$. If $f(x)$ is an odd function (meaning $f(-x) = -f(x)$), then $\int_{-a}^a f(x)dx = 0$. These symmetry properties can dramatically simplify calculations. The substitution method involves replacing $x$ with a function of a new variable $u$, along with adjusting the differential and limits. Integration by parts follows the formula $\int u \, dv = uv - \int v \, du$. For rational functions, partial fraction decomposition is often necessary. Trigonometric integrals may require identities like $\sin^2 x + \cos^2 x = 1$ or double angle formulas. Recognizing the type of function and choosing the appropriate technique is crucial for efficient problem-solving in JEE.

A function $f(x)$ is called even if $f(-x) = f(x)$ for all $x$ in its domain. Geometrically, the graph of an even function is symmetric about the y-axis. Examples include $f(x) = x^2$, $f(x) = \cos x$, and $f(x) = |x|$. A function $f(x)$ is called odd if $f(-x) = -f(x)$ for all $x$ in its domain. The graph of an odd function is symmetric about the origin. Examples include $f(x) = x^3$, $f(x) = \sin x$, and $f(x) = \tan x$. Most functions are neither even nor odd. However, any function can be expressed as the sum of an even function and an odd function: $f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) – f(-x)}{2}$, where the first term is even and the second is odd. Important properties for integration: if $f(x)$ is even, then $\int_{-a}^a f(x)dx = 2\int_0^a f(x)dx$. This halves the computational work. If $f(x)$ is odd, then $\int_{-a}^a f(x)dx = 0$ immediately, without any calculation. The product or quotient of two even functions is even. The product or quotient of two odd functions is even. The product of an even and an odd function is odd. The sum of two even functions is even, and the sum of two odd functions is odd. These properties extend to composition: if $f$ is even and $g$ is even, then $f \circ g$ is even. If both are odd, $f \circ g$ is odd. Understanding these symmetries helps in quickly evaluating integrals and solving equations in JEE problems.

A piecewise function is defined by different expressions over different parts of its domain. The greatest integer function $[x]$ is a classic example of a piecewise constant function. When integrating piecewise functions, we must split the integral at the boundaries where the function definition changes. For $f(x) = \begin{cases} f_1(x) & a \leq x < c \\ f_2(x) & c \leq x \leq b \end{cases}$, we have $\int_a^b f(x)dx = \int_a^c f_1(x)dx + \int_c^b f_2(x)dx$. The key steps are: identify all points where the function definition changes, determine the expression for $f(x)$ in each interval, integrate each piece separately using appropriate techniques, and sum all the results. Care must be taken with the endpoints to ensure continuity or proper handling of discontinuities. For the greatest integer function, the points of discontinuity are all integers, so we break the integral at integer boundaries within the integration limits. Within each interval $[n, n+1)$ where $n$ is an integer, $[x] = n$ is constant, making integration trivial. Common errors include forgetting to change limits when splitting integrals, incorrect handling of open vs. closed intervals at boundaries, and sign errors when dealing with negative values of $[x]$. Piecewise integration is fundamental in many JEE problems involving absolute values, modulus functions, and step functions used in physics and economics applications.

Mastering standard integration formulas is essential for quick problem-solving. Basic power rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. For $n = -1$: $\int \frac{1}{x} dx = \ln|x| + C$. Exponential: $\int e^x dx = e^x + C$ and $\int a^x dx = \frac{a^x}{\ln a} + C$. Trigonometric: $\int \sin x \, dx = -\cos x + C$, $\int \cos x \, dx = \sin x + C$, $\int \sec^2 x \, dx = \tan x + C$, $\int \csc^2 x \, dx = -\cot x + C$, $\int \sec x \tan x \, dx = \sec x + C$, $\int \csc x \cot x \, dx = -\csc x + C$. Inverse trigonometric: $\int \frac{1}{\sqrt{1-x^2}} dx = \sin^{-1} x + C$, $\int \frac{1}{1+x^2} dx = \tan^{-1} x + C$, $\int \frac{1}{x\sqrt{x^2-1}} dx = \sec^{-1} x + C$. Hyperbolic: $\int \sinh x \, dx = \cosh x + C$, $\int \cosh x \, dx = \sinh x + C$. For definite integrals, remember: $\int_a^b 1 \, dx = b – a$ (length of interval), $\int_0^a x \, dx = \frac{a^2}{2}$, $\int_0^{\pi} \sin x \, dx = 2$, $\int_0^{2\pi} \cos x \, dx = 0$. These formulas, combined with linearity and substitution, solve most standard integration problems efficiently. Regular practice and pattern recognition help in quickly identifying which formula or technique to apply.

Greatest Integer Function Piecewise Integration Even-Odd Properties Symmetric Limits Definite Integrals Breaking at Discontinuities