Question

If `int_((-pi)/4) ^(pi/4) x^3 * sin^4 x  dx` = k then k = ______.

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Answer 1

If `int_((-pi)/4) ^(pi/4) x^3 * sin^4 x  dx` = k then k = 0.

Explanation:

Let I = `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`

Let f(x) = `x^3 sin^4x`

∴ f( –x) = `(-x)^3 sin^4(- x)`

= `-x^3sin^4x`

= `-f(x)`

∴ f is an odd function.

∴ `int_((-pi)/4)^(pi/4) f(x)*dx = 0`,

i.e. `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx = 0`    ...(Property of def integration)

⇒ k = 0