Integrate a^x with respect to x

This comes up in C4 in A level maths and differentiating it could come up in C3. You can write a^x as exp(ln(a^x))=exp(xln(a)) then differentiating this, you get ln(a)exp(xln(a))=ln(a)a^x. By differentiating you can recognise the integral will be (a^x)/ln(a) +c or you can perform a u substitution where u=a^x then du=ln(a)a^xdx. dx=1/ln(a) * 1/u * du. Therefore the integral is now u/(u*ln(a)) du = 1/ln(a) du = u/ln(a) +c = a^x/ln(a) +c.

I have picked this since it could come up in C3 and C4 and I have had the same question asked to me by my peers before. The working can be further expanded by explaining how a^x can be written in terms of e and the natural logarithm, with these being inverse functions of each other, a topic within C3.