The value of ∫₋₁¹ x²e^([x³]) dx, where [ t ] denotes the greatest integer ≤ t, is :

Solution

Maths sir · 4 steps

Put t = x³ first. The x² dx is already sitting there, so the integral becomes plain step pieces.

1. Substitute. Let t = x³, so dt = 3x² dx. When x = -1, t = -1, and when x = 1, t = 1, so the limits stay the same.

   ∫₋₁¹ x² e^([x³]) dx = 1/3∫₋₁¹ e^([t]) dt

2. Split at zero. For -1 ≤ t < 0, [t] = -1, so the integrand is e⁻¹. For 0 ≤ t < 1, [t] = 0, so the integrand is e⁰ = 1. The single point t = 1 does not change the area.
3. Add the pieces. Each piece has width 1, so ∫₋₁¹ e^([t]) dt = 1/e + 1 = (e+1)/e.
4. So. The value is 1/3 × (e+1)/e = (e+1)/3e, which is option (d).

The idea to keep. With a greatest integer function, first substitute so the bracket holds a single variable. Then split the range at the integers, where the bracket is constant on each piece.

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