Q.Evaluate $\displaystyle\int \frac{dx}{\sqrt{x^{2}+25}}$.

A square root of $x^{2}+a^{2}$ in the denominator, with $a=5$, matches standard form 4 (a logarithmic result). …

Write the radicand as $x^{2}+5^{2}$, matching standard form 4 with $a=5$:

Check by differentiation: let $y=\log\left\lvert x+\sqrt{x^{2}+25}\right\rvert$. Then …

Reaching for the inverse-sine form (which is for $\sqrt{a^{2}-x^{2}}$, with the constant leading) when the radicand is actually $x^{2}+a^{2}$. Also mis-reading $25$ as $a=25$ instead of $a=5$; here $a$ only appears inside the root, so the value of $a$ d …