The value(s) of ∫₀¹ (x⁴(1 - x)⁴)/(1 + x²) dx is (are)
(a) 22/7 - π
(b) 2/105
(c) 0
(d) 71/15 - 3π/2

Q: Wait, why can't we just use substitution like $u = 1 + x^2$?

That is a great instinct! However, substitution works best when the numerator is a multiple of the derivative of the denominator (which would be $2x$). Here, our numerator is a high-degree polynomial ($x^8 - 4x^7 + 6x^6 - 4x^5 + x^4$). Because the degree of the numerator is much higher than the denominator, we must use polynomial long division first to reduce the integrand into a simpler form.

Q: Is there a faster way than long division? This looks like a lot of work.

While you could potentially use algebraic manipulation or synthetic division, polynomial long division is the most robust and reliable method for this type of problem. It guarantees you won't miss any terms. In JEE Advanced, accuracy is just as important as speed, and this systematic approach ensures you get the correct remainder every time.

Q: Why is the final answer $\frac{22}{7} - \pi$?

That is a beautiful observation! The value $\frac{22}{7}$ is a well-known rational approximation for $\pi$. The fact that our result is $\frac{22}{7} - \pi$ tells us that the area under this curve is a very small positive value, which aligns perfectly with the geometric reality of the function $y = \frac{x^4(1 - x)^4}{1 + x^2}$ being positive and small on the interval $[0, 1]$.

Question

The value(s) of $\int_0^1 \frac{x^4(1 - x)^4}{1 + x^2} dx$ is (are)

Options:

(A) $\frac{22}{7} - \pi$

(B) $\frac{2}{105}$

(C) 0

(D) $\frac{71}{15} - \frac{3\pi}{2}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are going to tackle a problem that might look like a nightmare of algebra at first glance, but it is actually a beautifully structured puzzle.

We are looking at the definite integral:

$$I = \int_0^1 \frac{x^4(1 - x)^4}{1 + x^2} dx$$

When you see a rational function where the degree of the numerator is significantly higher than the denominator, your first thought should always be: "How can I simplify this?" This is an improper rational function, and our goal is to break it down into manageable pieces.

### Phase 1: The Algebraic Expansion

Before we can integrate, we need to see what we are working with. The numerator is $x^4(1 - x)^4$.

Let's expand the binomial term $(1 - x)^4$ using the Binomial Theorem. It gives us $1 - 4x + 6x^2 - 4x^3 + x^4$.

Now, we multiply this by the $x^4$ sitting outside. Distributing that $x^4$ gives us the full numerator:

$$x^8 - 4x^7 + 6x^6 - 4x^5 + x^4$$

This is a degree-eight polynomial. Our denominator is just $x^2 + 1$. Since the degree of the numerator is greater than the degree of the denominator, we must perform polynomial long division.

### Phase 2: The Heavy Lifting

This is where the real work happens. We divide $x^8 - 4x^7 + 6x^6 - 4x^5 + x^4$ by $x^2 + 1$.

After careful division, we find that our rational function simplifies to a polynomial quotient plus a remainder term:

$$x^6 - 4x^5 + 5x^4 - 4x^2 + 4 - \frac{4}{x^2 + 1}$$

Look at that! We have transformed a terrifying fraction into a simple polynomial and a standard integral form.

### Phase 3: Integration and Evaluation

Now, we integrate term by term. The polynomial part is straightforward using the power rule $\int x^n dx = \frac{x^{n+1}}{n+1}$.

The last term, $-\int \frac{4}{x^2 + 1} dx$, is a classic standard integral that results in $-4 	an^{-1}(x)$. So, our antiderivative is:

$$F(x) = \frac{x^7}{7} - \frac{2x^6}{3} + x^5 - \frac{4x^3}{3} + 4x - 4	an^{-1}(x)$$

Now, we apply the limits from $0$ to $1$. At $x = 0$, everything vanishes.

At $x = 1$, we get:

$$\frac{1}{7} - \frac{2}{3} + 1 - \frac{4}{3} + 4 - 4	an^{-1}(1)$$

Since $	an^{-1}(1) = \frac{\pi}{4}$, the last term becomes $-4(\frac{\pi}{4}) = -\pi$.

Combining the numerical terms, we get $\frac{1}{7} - 2 + 5 - \pi$, which simplifies to:

$$\mathbf{\frac{22}{7} - \pi}$$

And there you have it—a perfect, elegant solution to a complex-looking problem. Keep practicing, and remember: every complex integral is just a series of simple steps waiting to be solved!

Animated Solution for Mathematics - Definite Integration: The value(s) of $\int_{0}^{1} \frac{x ^{4} ( 1 - x ) ^{4}}{1 + x ^{2}} d x$ is (are)

Select Answer:

Visualized Solution

The Definite Integral

Analyzing the Rational Function

Expanding $(1 - x)^{4}$

Multiplying by $x^{4}$

Polynomial Long Division Setup

Long Division: First Terms

Long Division: Final Terms

Rewriting the Integral

Integrating the Polynomial Part

Integrating the Remainder Term

Evaluating at Upper and Lower Limits

Substituting $tan^{- 1} (1)$

Combining the Numerical Terms

The Final Result

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

Phase 1

Phase 2

Phase 3

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Animated Solution for Mathematics - Definite Integration: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value(s) of ∫01​1+x2x4(1−x)4​dx is (are)