Problem 187 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of the integral \(I=\int_{0}^{1} x(1-x)^{n} d x\) is (A) \(\frac{1}{n+1}\) (B) \(\frac{1}{n+2}\) (C) \(\frac{1}{n+1}-\frac{1}{n+2}\) (D) \(\frac{1}{n+1}+\frac{1}{n+2}\)

Step-by-Step Solution

Answer

The value of the integral is \(\frac{1}{(n+1)(n+2)}\). Option C matches.

Step 1: Understand the Integral

We need to solve the integral \(I=\int_{0}^{1} x(1-x)^{n} dx\). This is a definite integral with the variable of integration being \(x\) from 0 to 1.

Step 2: Use Integration by Parts

We apply the method of integration by parts where \(u = x\) and \(dv = (1-x)^n dx\). Therefore, \(du = dx\) and to find \(v\), we integrate \(dv\):\[v = \int (1-x)^n dx.\]

Step 3: Integrate \(dv\)

To integrate \((1-x)^n\), we use the substitution \(t = 1-x\), hence \(dt = -dx\). The integral becomes:\[\int (1-x)^n dx = -\int t^n dt = -\frac{t^{n+1}}{n+1} + C = -\frac{(1-x)^{n+1}}{n+1}.\]

Step 4: Apply Integration by Parts Formula

With \(u = x\), \(du = dx\), and \(v = -\frac{(1-x)^{n+1}}{n+1}\), the integration by parts formula \(\int u \, dv = uv - \int v \, du\) gives:\[I = \left[-\frac{x(1-x)^{n+1}}{n+1}\right]_{0}^{1} + \int_0^1 \frac{(1-x)^{n+1}}{n+1} dx.\]

Step 5: Evaluate and Simplify the First Term

Evaluate \([-\frac{x(1-x)^{n+1}}{n+1}]_{0}^{1}\): At \(x = 1\), the term becomes 0, and at \(x = 0\), the term also becomes 0. Thus, the first part of the solution is 0.

Step 6: Simplify the Second Term

The second term is simplified as:\[\int_0^1 \frac{(1-x)^{n+1}}{n+1} dx = \frac{1}{n+1}\int_0^1 (1-x)^{n+1} dx.\]Now let \(t = 1-x\), then \(dt = -dx\), changing the limits gives \(t\) from 1 to 0:\[= \frac{1}{n+1} \int_1^0 t^{n+1} (-dt) = \frac{1}{n+1} \int_0^1 t^{n+1} dt.\]

Step 7: Integrate the Simplified Term

Continuing from the previous step, we integrate:\[= \frac{1}{n+1} \left[ \frac{t^{n+2}}{n+2} \right]_{0}^{1} = \frac{1}{(n+1)(n+2)} \times (1 - 0).\]

Step 8: Combine Results

The portions together reveal:\[I = 0 + \frac{1}{(n+1)(n+2)}.\]Adding the above, we find the solution.

Key Concepts

Integration by Parts

Integration by Parts is a method used to solve integrals that are products of functions. The formula is based on the product rule for differentiation and is given by:

• \(\int u \, dv = uv - \int v \, du\)

The key steps when using Integration by Parts involve:

• Identifying parts of the integrand to assign to \(u\) and \(dv\) where \(dv\) is easy to integrate.
• Computing \(du\) and \(v\) from the derivatives of \(u\) and integrating \(dv\).
• Filling them into the formula to simplify the original integral.

In our exercise, we chose \(u = x\) and \(dv = (1-x)^n dx\). The choice made it easier to compute \(du = dx\) and find \(v\) by integrating a slightly more complex function, which was tackled by applying the Substitution Method.

Substitution Method

The Substitution Method is often used to simplify integrals by transforming them into a more manageable form. The steps typically involve:

• Choosing a substitution that will simplify the integral.
• Changing the variable of integration from \(x\) to a new variable, such as \(t\).
• Replacing \(dx\) with \(dt\) using the derivative of the substitution.

In the given exercise, the substitution \(t = 1-x\) transformed our integral to one involving \(t^n\). This turned a complex polynomial expression into a power function, which is straightforward to integrate. After substituting and changing the limits of integration according to the new variable, the integral became easier to solve.

Mathematical Problem Solving

Mathematical Problem Solving in the context of definite integrals involves a strategic approach to break down complex problems and apply the right mathematical methods. The process usually follows these steps:

• Understanding the problem and identifying what is given and what needs to be found.
• Choosing an appropriate integration technique based on the form of the integral.
• Evaluating the integral, often by using limits to determine the specific value over an interval.
• Simplifying results to find a comprehensible solution.

In the exercise, integrating \(x(1-x)^n\) required the Integration by Parts method combined with substitution to handle the nested polynomial. After managing both components, the problem was simplified to a point where definite values could be substituted to obtain the final solution. This methodical and patient approach is central to solving mathematical problems effectively.