Apply integration by parts to evaluate integrals of products, including repeated application and the recovery of the original integral

What this dot point is asking

NESA wants you to integrate products of functions using integration by parts, the reverse of the product rule. You must choose the parts sensibly, apply the method more than once where needed, and handle the case where the original integral reappears and can be solved algebraically.

The formula

The product rule says $\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}$. Integrating both sides and rearranging gives integration by parts:

The aim is to swap the original integral for an easier one. Success depends entirely on a good split: $u$ should become simpler when differentiated, and $\dfrac{dv}{dx}$ should be something you can integrate.

Choosing the parts: LIATE

A reliable order of preference for choosing $u$ is LIATE:

• Logarithmic ($\ln x$)
• Inverse trigonometric ($\arctan x$)
• Algebraic ($x^n$)
• Trigonometric ($\sin x$, $\cos x$)
• Exponential ($e^x$)

Whatever appears earlier in this list is usually the better choice for $u$. For example, in $\int x\ln x\,dx$, the logarithm ranks above the algebraic term, so $u = \ln x$.

A first example

Evaluate $\int x e^x\,dx$. Let $u = x$ (algebraic) and $\dfrac{dv}{dx} = e^x$. Then $\dfrac{du}{dx} = 1$ and $v = e^x$. So

Repeated application

When the algebraic factor has a higher power, apply the method repeatedly. For $\int x^2 e^x\,dx$, first take $u = x^2$, $\dfrac{dv}{dx} = e^x$ to get $x^2 e^x - 2\int x e^x\,dx$, then reuse the result above. Each application lowers the power of $x$ by one until it vanishes.

The boomerang (recovery) case

For integrals such as $\int e^x \cos x\,dx$, two applications of integration by parts return a multiple of the original integral. You then solve algebraically. This is the "boomerang" because the integral comes back to you, letting you collect like terms and divide.