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Write question as 1/4 Integral of 4x(1+2x^2)^1/2
There is a rule which says as 4x is the derivative of the expression in the bracket then this part integrates to (1+2x^2)^3/2 divided by 3/2, that is
2/3(1+2x^2)^3/2
Hence complete answer is 1/4.2/3(1+2x^2)^3/2=1/6(1+2x^2)^3/2