Explain integration by parts with a worked example

Integration by parts is a powerful technique that helps integrate products of two functions. It comes from the product rule of differentiation and is used when the integrand is a product of two different types of functions (like polynomial × exponential, or log × algebraic). The rule is:

$$\int u\,dv = uv - \int v\,du$$

Here, u is the function you choose to differentiate (making it simpler), and dv is the part you integrate. The key is to pick u such that its derivative is simpler than itself, and dv such that its integral is manageable. A handy mnemonic to remember the order of preference for choosing u is ILATE rule: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. So choose u first from this list, and the remaining part becomes dv.

Let's work an example: $$\int x e^x\,dx$$. Here we have a product of algebraic ($x$) and exponential ($e^x$). By ILATE, Algebraic comes before Exponential, so choose u = x and dv = e^x dx. Then du = dx and v = e^x. Applying the formula:

$$\int x e^x\,dx = x e^x - \int e^x\,dx = x e^x - e^x + C$$

So the answer is $$e^x(x - 1) + C$$.

A common mistake is forgetting to include the constant of integration C at the end. Also, for harder cases like $$\int x^2 \sin x\,dx$$, you may need to apply integration by parts twice because the power of $x$ doesn't reduce to zero after one step. Practice choosing u wisely to make the integral simpler. Remember: the goal is to transform a difficult integral into a simpler one, and often the choice of u is the whole trick.