Explain integration by parts with a worked example

Integration by parts is the reverse of the product rule for differentiation. It helps integrate products of two functions. The formula is:

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

Here, choose $u$ (the part to differentiate) and $dv$ (the part to integrate). The goal is to make the new integral $\int v\,du$ simpler than the original. Use the ILATE rule to pick $u$: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential. Pick $u$ from the first category that appears in your product.

Steps: 1. Identify $u$ and $dv$. 2. Differentiate $u$ to get $du$. 3. Integrate $dv$ to get $v$. 4. Plug into the formula and simplify.

Worked example: Integrate $\int x\cos x\,dx$.

Here, product is $x$ (algebraic) and $\cos x$ (trigonometric). By ILATE, Algebraic comes before Trigonometric, so choose $u = x$, $dv = \cos x\,dx$.

Differentiate: $du = dx$.

Integrate: $v = \sin x$ (since $\frac{d}{dx}\sin x = \cos x$).

Now apply formula:

$$\int x\cos x\,dx = x\sin x - \int \sin x\,dx$$

Integrate $\sin x$: $\int \sin x\,dx = -\cos x$.

So the final answer is:

$$\int x\cos x\,dx = x\sin x + \cos x + C$$

(Don't forget the constant $C$!)

Key tip: If the product has a polynomial and an exponential, choose $u$ as the polynomial, because its derivative becomes simpler (degree reduces). For example, $\int x e^x\,dx$: pick $u=x$, $dv=e^x\,dx$, then $v=e^x$, $du=dx$, giving $x e^x - e^x + C$.

Common mistake: Choosing $u$ and $dv$ wrongly can make the integral harder. Always check that $\int v\,du$ is simpler. Practice with $\int \ln x\,dx$ (treat as $\ln x \cdot 1$, pick $u=\ln x$, $dv=1\,dx$) to get $x\ln x - x + C$.

Remember, integration by parts is your friend for products, and ILATE is your guide. Happy integrating!