Integral 1/(2+x^2) trig substitution. Let x=sqrt(2)*tan(theta) and use tangent secant identity.

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We compute the integral 1/(2+x^2) by applying trigonometric substitution. When we choose the trig substitution, we let x=sqrt(2)*tan(theta) so that when we square x, we can factor a 2 out of the denominator. We plan to use tangent secant identity, because we have the form "constant + variable squared" in the denominator.

After we make the 1/(2+x^2) trig substitution, we clean up the integral, guess the antiderivative, and substitute for theta in terms of x to obtain the final answer. We substitute for theta by solving the original substitution for theta, giving an inverse tangent of x/sqrt(2).