Q.Integrate the following function: $x^2 \log x$

Concept: Integration by Parts (using the LIATE rule — Logarithmic functions come first).

We set $u = \log x$ and $dv = x^2 \, dx$.

Then $du = \frac{1}{x} \, dx$ and $v = \frac{x^3}{3}$.

Applying the formula $\int u \, dv = uv - \int v \, du$:

Now integrate $x^2$: …

The key here is recognising that $\log x$ and $x^2$ are not related by a simple substitution — one is logarithmic, the other a power. When you have a product of two different kinds of functions, integration by parts is the natural tool. The formula is:

The art lies in choosing which part is $u$ and which is $dv$. A reliable rule of thumb for Indian exams is the ILATE order: Inverse, Logarithmic, Algebraic, Trigonometric, Exponential. Whichever function comes first in ILATE should be taken as $u$ (the one you differentiate). Here, $\log x$ is Logarithmic, and $x^2$ is Algebraic. So $\log x$ comes first — it becomes $u$, and $x^2$ becomes $dv$.

Let’s work through it.

1. Set up the parts.

   Let $u = \log x$ and $dv = x^2 \, dx$.

   Differentiate $u$: $du = \frac{1}{x} \, dx$.

   Integrate $dv$: $v = \int x^2 \, dx = \frac{x^3}{3}$.

2. Apply the integration by parts formula.

Simplify the new integral: $\frac{x^3}{3} \cdot \frac{1}{x} = \frac{x^2}{3}$.

3. Integrate the simpler term.

4. Combine everything. …

Method: Integration by parts on a power times a logarithm

Use this whenever the integrand is $x^{n}\log x$ (a power multiplied by a logarithm) — two unrelated function types that no substitution links.

Steps

Step 1: Recognise the product type and pick $u$ by ILATE.

The factors are Logarithmic ($\log x$) and Algebraic ($x^{n}$). In the ILATE order (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential) the logarithm comes first, so it is the part you differentiate:

Log is chosen as $u$ because it has no elementary integral on its own, but its derivative $1/x$ is simple.

Step 2: Build $du$ and $v$.

Step 3: Apply the by-parts formula.

$\int u\,dv=uv-\int v\,du=\frac{x^{n+1}}{n+1}\log x-\int\frac{x^{n+1}}{n+1}\cdot\frac1x\,dx.$ …

Common Mistakes

Mistake 1: Choosing $u=x^{2}$ and $dv=\log x\,dx$.

Why it's wrong: that forces you to integrate $\log x$ (which itself needs by parts), making the problem harder, not easier. Correct approach: by ILATE the logarithm is $u$ — differentiate $\log x$, integrate the power.

Mistake 2: Simplifying $\dfrac{x^{3}}{3}\cdot\dfrac1x$ incorrectly. …