3 min readβ’april 26, 2020

Catherine Liu

**Enduring Understanding FUN-6:**Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.Β**Essential Knowledge FUN-6.E.1:**Integration by parts is a technique for finding antiderivatives.

Whenever thereβs an integral where **the integrand** (the thing being integrated) **is a product of two functions**, one of the functions will be f(x) and the other will be gβ(x). If other methods like u-substitution donβt work, **try integration by parts. π―**

Sometimes, you might see the equation written like this:

One of the functions will be **U** and the other will be **V**. However, we still need to know what dV and dU actually mean. If U is a function of x, then U = f(x). Taking the derivative of both sides, we find that dU = fβ(x)dx. Same thing for V: If V = g(x),Β then dV = gβ(x)dx. This new way of writing the integration by parts formula is just a **sneakier version of the original formula.**** π€**

The best way to explain this formula is through an example. Take the following integral:

At first glance, you may try u-substitution, but this method will take you nowhere. Instead, notice that the integrand is a product of two functions. This is an opportunity to use integration by parts. Letβs recall the formula:

The first step should be to figure out what f(x) and gβ(x) should be. This step is probably the most challenging of this method, because you have to choose which is which in order to make the problem work.Β

This is where a handy acronym comes into play: **L.I.A.T.E. **It stands for **L**ogarithmic, **I**nverse trigonometric, **A**lgebraic, **T**rigonometric, and **E**xponential.

The function closer to the βEβ side should be gβ(x) and the function closer to the βLβ side should be f(x).Β

For our example problem, that means:

We know we need g(x) and fβ(x) to use the integration by parts formula, so letβs find those now:

- Check to make sure that
**u-substitution and other methods donβt work**. - Check to make sure that the
**integrand is a product of two functions**. **Use L.I.A.T.E.**to pick which function should be f(x) and which should be gβ(x).- Plug into the formula and
**integrate**.

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