# 11.4 Integral arithmetic

This technique goes by different names, but integral arithmetic captures the basic idea pretty well; sometimes it is very helpful to treat integrals as algebraic objects in order to find their value.

A very common example of this is where, by integrating $f(x)$ (or any other integrable function) with respect to $x$, we can arrive with an equation of the form (here we define $k$ to stand for \sayan integral we know to directly find the value of)

It is important that $a\neq 1$ (because if $a$ is equal to one then we cannot solve for $\int f(x)dx$), in which case we can just subtract $a\int f(x)dx$ from both sides, to solve for $\int f(x)dx$.

###### Example 11.4.1

Find the value of

Solution: Start by integrating by parts (as in Section 11.2)

Then integrate $\int\frac{e^{2x}}{2}[-\sin(x)]$ by parts.

Overall then, we have

And we can add $\int\frac{e^{2x}}{2}\cos(x)dx$ to both sides, giving that

and then after multiplying both sides by $\frac{4}{5}$, we get that

Integrating by parts can get really messy - good presentation is key.