improper integral [x^2/(9+x^6)dx] x= -infinite  to infinite show steps by using the difinition



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improper integral [x^2/(9+x^6)dx], x = -infinity  to x = infinity.

x^2/(9+x^6) = x^2/(9+(x^2)^3) is an even function. Therefore, the given integral is equal to  2*integral (x^2/(9+x^3)) dx, x =0 to infinity.

Let us have a transformation x^3+9 = t. Then,

x^2dx = (1/3)dt. When x= 0, t =9 and x= infinity, t = infinity.

Therefore, the given integral = 2Integral {1/t}(1/3)dt t= 9 to t= inf.

=(2logt), t = 0 to t=inf.

= 2 log (infinity) - 2log9

= infinity.

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