Page:Calculus Made Easy.pdf/220

 the problem how to integrate $$x^{-1}\, dx$$. Indeed it should be frankly admitted that this is one of the curious features of the integral calculus:–that you can’t integrate anything before the reverse process of differentiating something else has yielded that expression which you want to integrate. No one, even to-day, is able to find the general integral of the expression,

because $$a-x^2$$ has never yet been found to result from differentiating anything else.

Another simple case.

Find $$\int (x + 1)(x + 2)\, dx$$.

On looking at the function to be integrated, you remark that it is the product of two different functions of $$x$$. You could, you think, integrate $$(x + 1)\, dx$$ by itself, or $$(x + 2)\, dx$$ by itself. Of course you could. But what to do with a product? None of the differentiations you have learned have yielded you for the differential coefficient a product like this. Failing such, the simplest thing is to multiply up the two functions, and then integrate. This gives us

And this is the same as

And performing the integrations, we get