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Question
If `int_((-pi)/4) ^(pi/4) x^3 * sin^4 x dx` = k then k = ______.
Options
1
2
4
0
Solution
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
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