Evaluate: ∫x3 sin2 x dx for x ∈ [-π/4,π/4] – InterviewSolution

1.

Evaluate: ∫x3 sin2 x dx for x ∈ [-π/4,π/4]

Answer»

Let f(x) = x³ sin² x = x³ (sin x)²

∴ f(- x) = (- x)³ (sin (- x))² = (- x)³ (- sin x)²

= – x³ sin² x = -f(x)

f(-x) = -f(x)

∴ f(x) is an odd function.

∴ ∫x³ sin² x dx for x ∈ [-π/4,π/4] = 0 (by property)

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