Q.(b) Evaluate : ∫ 𝒙(𝟏 − 𝒙)𝒏𝒅𝒙; (𝐰𝐡𝐞𝐫𝐞 𝒏 ∈ 𝑵). 𝟏 𝟎

The Beta Function: A First Look

Think of the Beta function as a tool that measures how two quantities blend together in a certain integral. You've seen the Gamma function $\Gamma(n) = (n-1)!$ as a generalization of factorial. The Beta function is like a "product" of two Gamma functions, appearing in its own right in probabilities, integrals of powers, and combinatorial identities.

Intuition: Why "Beta"?

Given two positive numbers $a$ and $b$, the Beta function $B(a,b)$ answers: if I integrate $t^{a-1}(1-t)^{b-1}$ from $t=0$ to $t=1$, what value do I get?

The integrand is a product of two powers — one grows near $t=0$ (if $a>1$), the other near $t=1$ (if $b>1$). The integral "balances" these influences. This shape appears in statistics (the Beta distribution), physics (phase space integrals), and combinatorics (binomial coefficients).

Precise Definition

For any two positive reals $a > 0$, $b > 0$:

The integral converges because near $t=0$, $t^{a-1}$ is integrable when $a>0$, and near $t=1$, $(1-t)^{b-1}$ is integrable when $b>0$.

Key Properties (Memorize These)

1. Symmetry:

(Substitute $u = 1-t$ in the integral.)

2. Relation to Gamma function (the most important formula):

For positive integers $m,n$:

3. Special values:

   • $B(1,1) = 1$
   • $B\left(\frac12, \frac12\right) = \pi$ (since $\Gamma(1/2)=\sqrt{\pi}$, $B(1/2,1/2) = \frac{\sqrt{\pi}\cdot\sqrt{\pi}}{\Gamma(1)} = \pi$)

4. Trigonometric form (substitute $t = \sin^2\theta$):

How to Use It: A Simple Example

Problem: Evaluate $\int_{0}^{1} x^3 (1-x)^2 \, dx$.

Solution: This is $B(4,3)$ because $a-1=3 \Rightarrow a=4$, $b-1=2 \Rightarrow b=3$.

Common Mistake to Avoid

Why It Matters for Exams

In Indian competitive exams (JEE, GATE, etc.), the Beta function appears as direct evaluation of $\int_0^1 x^m (1-x)^n dx$ and as a shortcut to compute definite integrals via Gamma functions. Once you express an integral as a Beta, rewrite it as a ratio of Gamma functions and use known values ($\Gamma(n+1)=n!$, $\Gamma(1/2)=\sqrt{\pi}$) for a numeric answer.

Final takeaway: The Beta function is just a compact name for a specific integral that appears repeatedly. Learn its Gamma connection, and you'll solve many integrals in seconds.

The Beta function goes beyond the standard NCERT CBSE Class 11/12 syllabus, but it directly extends the Gamma-function and definite-integral ideas that appear at the edges of the Class 12 Integrals chapter, and it's a genuinely useful shortcut for JEE Advanced and engineering-entrance level integral problems. Students searching 'beta function formula and examples' or 'relation between beta and gamma function' will find this B(a,b) = Γ(a)Γ(b)/Γ(a+b) identity is exactly the result those references point to.