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प्रश्न
Evaluate the following : `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`
उत्तर
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.
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