Exdexcel Formula Sheet lays Pooh Traps for Further Mathematics Pupils

FP3 integration standard forms could lead unwary pupils and teachers into a Pooh Trap. The following issue arises each year as the finer points of this course get sharpened.

When consulting the Edexcel formula sheet to find the integral,

$\displaystyle \int \dfrac{1}{\sqrt{a^{2}+x^{2}}} \textnormal{d}x$

two anti-derivatives are given,

$\textnormal{arsinh}\left(\dfrac{x}{a}\right)\ \ \ \textnormal{and}\ \ \ \ln\{x+\sqrt{x^{2}+a^{2}}\}$

Some might be tempted to infer that these functions are equal, this is the Pooh Trap, because whilst they both differentiate to $\dfrac{1}{\sqrt{a^{2}+x^{2}}}$ they are not equal.

Play with exponential functions and quadratic equations give us the logarithmic form of $\textnormal{arsinh}\ x$ which is also represented in formula books.

$\textnormal{arsinh}\ x=\ln\{x+\sqrt{x^{2}+1}\}$

Then,
$\textnormal{arsinh} \left(\dfrac{x}{a}\right)=\ln\left\{\dfrac{x}{a}+\sqrt{\left(\dfrac{x}{a}\right)^{2}+1}\right\}=\\\ln\left\{\dfrac{x+\sqrt{x^{2}+a^{2}}}{a}\right\}=\ln\left\{x+\sqrt{x^{2}+a^{2}}\right\}-\ln a$
The two anti-derivatives differ by the constant $\ln a$.

In a definite integration, this constant is added and then taken away making no difference as to which anti-derivative the student uses. In a particular solution to a differential equation, the evaluation of the integral using boundary condition would lead to two different constant values.

The worse case is that a student takes the integration standard from and reads it as the logarithmic form of the inverse of $\sinh x$. I have seen this happen.