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

Solution

Maths sir · 4 steps

Split the fraction first: one piece cancels completely, and King's rule handles the other.

1. Split the integrand. Write g(x) = √(|x|-x). The numerator is eˣ + g(x)(eˣ + e⁻ˣ), so the integrand becomes eˣ/(eˣ+e⁻ˣ) + g(x).
2. King's property. Let I = ∫₋₁¹ eˣ/(eˣ+e⁻ˣ) dx. Replacing x by -x gives I = ∫₋₁¹ e⁻ˣ/(e⁻ˣ+eˣ) dx. Adding the two forms gives 2I = ∫₋₁¹ 1 dx = 2, so I = 1.
3. The root term. For x ≥ 0, |x| - x = 0, so g(x) = 0. For x < 0, |x| - x = -2x, so only the interval [-1, 0] contributes.

   ∫₋₁⁰√-2x dx = √2∫₀¹√t dt = √2×2/3 = 2√2/3

4. So. The value is 1 + 2√2/3, which is option (a).

The idea to keep. When |x| sits inside a function, split the interval at 0 and find where it vanishes. On a symmetric interval, add f(x) and f(-x) to make eˣ and e⁻ˣ fractions collapse to 1.

Answer from the NTA final answer key. Solution by PrepWiser.