Q.Evaluate: $\int \frac{1}{7+2x^2}\,dx$

A general quadratic denominator (or quadratic under a square root) is reduced to one of the seven special integrals by a five-step routine: pull the leading coefficient $a$ (or $\sqrt a$) outside; complete the square on the remaining $x^2+\frac bax$ by adding and subtracting $\left(\frac12\cdot\text{coefficient of }x\right)^2$; write the result as a sum or difference of two squares, $(x+\beta)^2\pm\alpha^2$; shift the standard formula (since $\int f(x)\,dx=g(x)+c\Rightarrow\int f(x+\beta)\,dx=g(x+\beta)+c$); and finally apply whichever of the seven special formulae matches. Whether the final answer is an inverse tangent, a logarithm, or an inverse sine depends entirely on the sign that appears after completing the square — a positive constant next to the shifted-square term under a plain denominator gives $\tan^{-1}$; a difference of two real squares gives a log; a square root with a positive constant minus the shifted square gives $\sin^{-1}$.