The value of ∫₋₁¹ ((1+√(|x|-x))eˣ + (√(|x|-x))e⁻ˣ) / (eˣ + e⁻ˣ) dx is equal to

Q: Why did we use the $\int_{-a}^a$ property? Can't we just integrate directly?

You could try, but you'd have to split the integral at $x=0$ manually. The property $\int_{-a}^a f(x) dx = \int_0^a (f(x) + f(-x)) dx$ is a powerful tool that handles the absolute value discontinuity at $x=0$ automatically. It's not just a shortcut; it's a strategic maneuver to keep your algebra clean.

Q: I'm confused about $|-x|$. If $x$ is positive, is $|-x|$ equal to $x$ or $-x$?

Great question! If $x > 0$, then $-x$ is negative. The definition of absolute value is $|u| = u$ if $u \ge 0$ and $-u$ if $u < 0$. So, $|-x| = -(-x) = x$. Always test with a number, like $x=0.5$. Then $|-0.5| = 0.5$. It works!

Q: Why did the $e^x + e^{-x}$ terms cancel out so perfectly? Is that a coincidence?

In JEE Advanced problems, nothing is a coincidence! The problem is designed with 'structural symmetry.' The exponential terms are there to distract you, but they are also the 'glue' that holds the expression together. When you see complex expressions like this, trust that there is a hidden factorization waiting to happen.

Question

The value of $\int_{-1}^1 \frac{(1+\sqrt{|x|-x})e^x + (\sqrt{|x|-x})e^{-x}}{e^x + e^{-x}} dx$ is equal to

Options:

(A) $3 - \frac{2\sqrt{2}}{3}$

(B) $2 + \frac{2\sqrt{2}}{3}$

(C) $1 - \frac{2\sqrt{2}}{3}$

(D) $1 + \frac{2\sqrt{2}}{3}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The problem presents an integral from $-1$ to $1$ involving exponential functions, absolute values, and square roots. While it appears intimidating, we can simplify it using symmetry properties.

The integral is defined as:

$$I = \int_{-1}^{1} f(x) dx$$

### The Symmetry Weapon

Whenever you encounter limits from $-a$ to $a$, you should immediately consider the property:

$$\int_{-a}^{a} f(x) dx = \int_{0}^{a} (f(x) + f(-x)) dx$$

This property is powerful because it allows us to handle the absolute value function $|x|$ by splitting the domain into positive and negative regions. By applying this, we effectively halve our workload and set the stage for simplification.

### Taming the Absolute Value

Let us evaluate $f(x)$ for $x \in (0, 1]$. Since $x$ is positive, $|x| = x$. The term inside the square root becomes $\sqrt{x-x} = 0$.

Thus, the function simplifies to:

$$f(x) = \frac{e^x}{e^x + e^{-x}}$$

Now, consider $f(-x)$. Replacing $x$ with $-x$, the term inside the square root becomes $|-x| - (-x)$. Since $x > 0$, then $|-x| = x$, resulting in $\sqrt{x+x} = \sqrt{2x}$.

Consequently, $f(-x)$ becomes:

$$f(-x) = \frac{(1+\sqrt{2x})e^{-x} + \sqrt{2x}e^x}{e^{-x} + e^x}$$

### The Algebraic Miracle

We now add $f(x)$ and $f(-x)$. Notice that both fractions share the common denominator $e^x + e^{-x}$.

Summing the numerators, we obtain:

$$e^x + (1+\sqrt{2x})e^{-x} + \sqrt{2x}e^x$$

Grouping the terms yields:

$$(e^x + e^{-x}) + \sqrt{2x}(e^{-x} + e^x) = (e^x + e^{-x})(1 + \sqrt{2x})$$

Dividing by the denominator, the exponential terms cancel out completely. We are left with the remarkably simple integrand:

$$1 + \sqrt{2x}$$

### Final Calculation

We have reduced the expression to the integral of $1 + \sqrt{2x}$ from $0$ to $1$:

$$I = \int_{0}^{1} (1 + \sqrt{2}x^{1/2}) dx$$

Applying the power rule for integration:

$$I = \left[ x + \sqrt{2} \cdot \frac{x^{3/2}}{3/2} \right]_{0}^{1} = \left[ x + \frac{2\sqrt{2}}{3}x^{3/2} \right]_{0}^{1}$$

Evaluating at the limits, we reach the final result:

**$$I = 1 + \frac{2\sqrt{2}}{3}$$**

Animated Solution for Mathematics - Definite Integration: The value of $\int_{- 1}^{1} \frac{( 1 + ∣ x ∣ - x ) e ^{x} + ( ∣ x ∣ - x ) e ^{- x}}{e ^{x} + e ^{- x}} d x$ is equal to

Select Answer:

Visualized Solution

Given Integral

Definite Integral Property

Evaluating $f (x)$ for $x > 0$

Setting up $f (- x)$

Simplifying $f (- x)$

Adding the Functions

Factoring the Numerator

Simplified Integrand

Visualizing the Integral

Performing the Integration

Applying Limits

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

The Symmetry Weapon

Taming the Absolute Value

The Algebraic Miracle

Final Calculation

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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 of ∫−11​ex+e−x(1+∣x∣−x​)ex+(∣x∣−x​)e−x​dx is equal to