Animated Solution for Mathematics - Definite Integration: The value of the integral $I={\int }_0^{1}x(1-x{)}^{n}dx$ is

Mastering the Definite Integral

Welcome, future engineer! Today, we are going to tackle a problem that perfectly illustrates the difference between "brute-forcing" a solution and "solving" it with mathematical intuition.

We are looking at the integral $I={\int }_0^{1}x(1-x{)}^{n}dx$. At first glance, it looks like a standard calculus problem, but it hides a beautiful secret that separates the top-tier students from the rest.

When you first see $(1-x{)}^n$, your instinct might be to reach for the Binomial Theorem. You might think, "I can just expand this!"

If you did, you would be looking at:

While mathematically correct, this path is a labyrinth. You would end up with a long, complex series of terms to integrate individually.

In the high-pressure environment of the JEE Advanced, time is your most precious resource. We need a smarter, more elegant approach.

A Geometric Revelation

This is where the "King's Property" comes into play. It is one of the most powerful tools in your JEE arsenal.

The property states that for any continuous function $f(x)$, the integral from $0$ to $a$ is invariant under the transformation $x\to a-x$:

Geometrically, this is stunning. It tells us that if you flip the graph of the function horizontally across the midpoint of the interval $[0,a]$, the area under the curve remains exactly the same.

It is a symmetry trick that turns complex problems into simple ones.

Let us apply this to our integral where $a=1$. We replace every instance of $x$ with $(1-x)$:

Now, look closely at the term inside the curly braces. We have $1-(1-x)$. The ones cancel out, and the negative signs combine to give us $x$.

The entire expression collapses beautifully into:

Do you see the magic? We have transformed a product involving a binomial power into a simple polynomial. By distributing $x^n$ into the bracket, we get:

Now, we are on familiar ground. We can integrate these terms using the standard power rule, $\int x^{k}dx=\frac{x^{k+1}}{k+1}$.

Applying this to our expression, we get:

Evaluating this at the upper limit $x=1$ gives us $\frac{1}{n+1}-\frac{1}{n+2}$. When we evaluate at the lower limit $x=0$, both terms vanish.

Thus, our final result is:

Take a moment to appreciate what just happened. We avoided a massive binomial expansion and arrived at the answer using nothing but the inherent symmetry of the interval.

This is the essence of JEE mathematics: finding the path of least resistance through deep conceptual understanding. Keep this "King's Property" in your toolkit, and you will find that even the most intimidating integrals have a hidden, elegant solution waiting to be discovered.