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Question
Evaluate the following : `int_(-a)^(a) (x + x^3)/(16 - x^2)*dx`
Solution
Let I = `int_(-a)^(a) (x + x^3)/(16 - x^2)*dx`
Let f(x) = `(x + x^3)/(16 - x^2)`
∴ f( –x) = `((-x)+(-x)^3)/(16 - (- x)^2`
= `(-(x+x^3))/(16 - x^2)`
= `-f(x)`
∴ f is an odd function.
∴ `int_-a^a f(x)*dx = 0, "i.e." int_a^a (x + x^3)/(16 - x^2)*dx` = 0.
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