
We learn a new technique, called integration by parts, to help find antiderivatives of certain types of products by reexamining the product rule for differentiation.

We have seen applications of integration such as finding areas between curves, calculating volumes of certain solids, and some physical applications. In order to compute these definite integrals we have relied on the Fundamental Theorem of Calculus, which allows us to compute the definite integral easily provided that we can find an antiderivative of the integrand. In some cases, these applications can lead us to antiderivatives that cannot be evaluated using the basic techniques discussed so far.

Consider the following example.

### The integration by parts formula

The technique we applied in the above example extends to many other products of functions. We summarize the result of the argument in the preceding example first, then study other types of integrals for which this is a useful technique:

In general, we want to choose $u$ so the resulting integral on the righthand side is “easier” to compute, but in doing so, we must be able to integrate our choice for $\d v$.

We’ll now work some standard examples to develop some intuition for the technique.

Sometimes, we have “disguised products” in the integrand, as in the following example.

Sometimes, several techniques arise in the computation of antiderivatives.

### Repeated integration by parts

The integration by parts formula is intended to replace the original integral with one that is easier to determine. However the integral $\int v \d u$ that results may also require integration by parts. This can lead to situations where we may need to apply integration by parts repeatedly until we obtain an integral which we know how to compute.

Sometimes, we can perform integration by parts a few times and recover our original integral.

Sometimes we can use integration by parts to give a reduction formula.

This reduction formula shows how to ”reduce” the power of $x$ so the resulting integral is slightly easier. We may use repeated applications of this rule to keep reducing the power of $x$ until we arrive at an integral we know how to compute directly.

### Final thoughts

When we covered the substitution method for antiderivatives, we saw that there was no fixed procedure for choosing $u$. There were only certain rules of thumb that might guide you to better or worse choices of which part of the integrand to substitute. The same idea applies in integration by parts. There is no procedure that tells you the best choice for $u$ and $\d v$. However here is a useful heuristic (“rule of thumb”) that can guide your choice.

The heuristic is referred by the mnemonic ILATE. The individual letters stand for different types of functions:

The idea is that when choosing $u$ and $\d v$, one looks at the types of functions that show up in the integrand. Function types that occur earlier in ILATE are better choices for $u$ while those that appear late are better choices for $\d v$. This is because functions near the top of ILATE generally become simpler when they are differentiated.

For example, in $\int x \ln (x) \d x$, the integrand is made up of a product of a algebraic function $x$ and a logarithmic function $\ln (x)$. ILATE would suggest choosing $u=\ln (x)$ since logarithmic comes above algebraic in ILATE. Thus $\d v=x \d x$ since that is the remaining portion. However one should keep in mind that ILATE is simply a rule of thumb that does not always apply and can actually make the problem more difficult to solve in some instances.

“Mathematics is the science of skillful operations with concepts and rules invented for this purpose” - Eugene Wigner