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Note that the derivative of an odd function is an even function from which it follows that the integral of an even function is an odd function plus some constant C. $$ Notice that $\sqrt{2+x^2}\sin^2 x$ is an even function. Thus, $f(x)=3x^2+oddf(x)+C$ . Using the fact that $f(0)=5$ yields $C=5$ . Solving now for the integral $\int\limits_{-2}^2{f(x)\ dx}=\int\limits_{-2}^2{(3x^2+odd f(x)+5)\ dx}=36$ *Note that $\int\limits_{-a}^a{oddf(x)\ dx}=0$