Problem 63 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\int_{0}^{\infty} e^{-a x} d x=\frac{1}{a}\), then \(\int_{0}^{\infty} x^{n} e^{-a x} d x\) is (A) \(\frac{(-1)^{n} n !}{a^{n+1}}\) (B) \(\frac{(-1)^{n}(n-1) !}{a^{n}}\) (C) \(\frac{n !}{a^{n+1}}\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (C) \( \frac{n!}{a^{n+1}} \).

Step 1: Identify Formula to Use

We need to identify the integration technique required here. Based on the form given in the problem, the integral is of the form of the Gamma function. Recall the formula \( \Gamma(n+1) = \int_0^\infty x^n e^{-x} \, dx = n! \). However, here we need to modify it for \( e^{-ax} \).

Step 2: Variable Substitution

To match the formula for the Gamma function, perform a substitution. Let \( u = ax \), thus \( du = a \, dx \) or \( dx = \frac{du}{a} \). This changes the integral from \( \int_{0}^{ o \infty} x^{n} e^{-ax} \, dx \) to \( \int_{0}^{\infty} (\frac{u}{a})^n e^{-u} \, \frac{du}{a} \).

Step 3: Simplify the Integral

This substitution simplifies the integrals to \( \frac{1}{a^{n+1}} \int_{0}^{\infty} u^n e^{-u} \, du \). Recognize this as the definition of the Gamma function \( \Gamma(n+1) = n! \).

Step 4: Solve the Integral

The integral is now in the form \( \int_{0}^{\infty} u^n e^{-u} \, du = \Gamma(n+1) = n! \). Substitute this result back into our expression: \( \frac{1}{a^{n+1}} \times n! \).

Step 5: Identify Correct Answer

From the simplification, \( \int_{0}^{\infty} x^n e^{-ax} \, dx = \frac{n!}{a^{n+1}} \). Compare with given options, it matches option (C).

Key Concepts

Gamma Function

The Gamma function, denoted as \( \Gamma(n+1) \), is an extension of the factorial function to complex numbers. It is defined by the integral:\[\Gamma(n+1) = \int_0^\infty x^n e^{-x} \, dx = n!\]This definition is crucial when dealing with integrals involving the exponential function and power of \( x \). The Gamma function helps in solving these integrals by connecting them to the factorials we use frequently in combinatorics and other areas of mathematics.
The integral will yield a factorial of \( n \), which can greatly simplify many problems. It turns the problem into an evaluation of a constant rather than a function of \( x \). Understanding

• how the Gamma function generalizes the factorial notation
• and how it applies in various fields

can provide deeper insight into mathematical integration techniques.

Variable Substitution

Variable substitution is a powerful technique used to simplify complex integrals. In this exercise, we perform substitution by letting \( u = ax \). Consequently, this changes our differential from \( dx \) to \( \frac{du}{a} \), simplifying the integral expression.
Substitution enables us to convert the original function into one that's recognizable and easier to evaluate. This especially applies when matching forms, like transitioning into the Gamma function form:

• Convert \( x^n e^{-ax} \) to \( \left(\frac{u}{a}\right)^n e^{-u} \)
• The limits of integration remain from \( 0 \) to \( \infty \) since \( u = ax \)

Consequently, this approach reduces complexity and highlights the expression's underlying structure. Through substitution, complicated integrals become more manageable and lead to elegant solutions.

Definite Integral

A definite integral, denoted \( \int_{a}^{b} f(x) \, dx \), calculates the area under a curve within specified bounds, \( a \) to \( b \). In the context of this problem, the integrals have bounds from \( 0 \) to \( \infty \). Definite integrals are foundational in evaluating expressions over continuous intervals.
When calculating a definite integral, the function is evaluated at both endpoints, requiring:

• Substitution using limits \( 0 \) and \( \infty \)
• Recognizing the convergence of the integral

Definite integrals extend beyond mere calculations of area—offering solutions to complex equations linking numerous concepts, such as exponential decay in probability and physics.

Factorial Function

The factorial function, denoted as \( n! \) (read "n factorial"), is a product of all positive integers up to \( n \). It is defined as:\[n! = n \times (n-1) \times (n-2) \times \cdots \times 1\]Factorials grow extremely large quite rapidly. They are used in permutations, combinations, and series expansions across various branches of mathematics.
In this problem, the Gamma function transforms to \( n! \), reinforcing how integrals over an exponential function and its power reduce to a factorial expression. Factorials bring a simple, powerful tool for computation, making otherwise intimidating mathematical problems more tractable.