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Find the value of`int_(-pi/3)^(pi/3)(pi+4x^3)/(2-cos(|x|pi/3))dx` |
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Answer» Let `I=int_(-pi//3)^(pi//3) (pi+4x^(3))/(2-cos(|x|+(pi)/3))dx+int_(-pi//3)^(pi/3)(4x^(3))/(2-cos(|x|+(pi)/3))dx` The second integral becomes zero as integrand being an odd function of `x`. `:. I=2pi int_(0)^(pi//3)(dx)/(2-cos(x+(pi)/3))` Let `x+pi//3=y` or `dx=dy`. Also as `xto0,y to pi//3`, and as `x to pi//3, yto2pi//3`. `:. I=2pi int_(pi//3)^(2pi//3)(dy)/(2-cosy)` `=2pi int_(pi//3)^(2pi//3) (dy)/(2-(1-tan^(2)y//2)/(1+tan^(2)y//2))` `=2pi int_(pi//3)^(2pi//3)(1/2sec^(2)y//2)/(tan^(2)y//2+(1//sqrt(3))^(2))dy` `=(4pisqrt(3))/3[tan^(-1)(sqrt(3) tan y//2)]_(pi//3)^(2pi//3)` `=(4pi)/(sqrt(3))[tan^(-1)3-tan^(-1)1]` `=(4pi)/(sqrt(3))[tan^(-1)3-pi//4]` |
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