Q.Integrate the following function: $\frac{1}{x+x \log x}$

U Substitution: The Reverse Chain Rule

The chain rule differentiates composite functions: the derivative of $\sin(x^2)$ is $\cos(x^2) \cdot 2x$ — differentiate the outer function, then multiply by the derivative of the inside. Integration asks the reverse: given $\cos(x^2) \cdot 2x$, find the original function. That's what u substitution does — it reverses the chain rule.

The Core Intuition

When an integral looks like "a function times the derivative of its inside," substitute the inside with $u$ and the derivative of the inside with $du$. Consider:

Here $2x$ is the derivative of $x^2$, and $x^2$ is the inside of $\cos(x^2)$. Let $u = x^2$, so $du = 2x \, dx$:

Check: the derivative of $\sin(x^2)$ is $\cos(x^2) \cdot 2x$.

The Precise Statement

Valid provided $g$ is differentiable and the resulting integral in $u$ is simpler.

The Step-by-Step Method

1. Identify a function $g(x)$ whose derivative $g'(x)$ also appears (possibly up to a constant factor).
2. Set $u = g(x)$, compute $du = g'(x) \, dx$.
3. Rewrite the entire integral in $u$ and $du$ — every $x$ and $dx$ must be replaced.
4. Integrate with respect to $u$.
5. Substitute back $u = g(x)$.

A Second Example (with a constant factor)

Evaluate $\int x \sqrt{x^2 + 1} \, dx$. Let $u = x^2 + 1$, so $x \, dx = \frac{1}{2} du$:

When Does It Work?

When the integrand is something times the derivative of something inside. Common patterns:

• $x \cdot f(x^2)$ — derivative of $x^2$ is $2x$, so $u = x^2$
• $e^{g(x)} \cdot g'(x)$ — derivative of $g(x)$ appears
• $\frac{g'(x)}{g(x)}$ — leads to $\log|g(x)|$

The Definite Integral Case

Either change the limits (when $x = a$, $u = g(a)$; when $x = b$, $u = g(b)$; then integrate in $u$), or integrate in $u$, substitute back, and use the original limits. Changing limits is cleaner:

Common Mistake to Avoid

Don't confuse $du$ with $\Delta u$. $du$ is a differential — the exact relationship $du = g'(x) dx$ that holds inside the integral. Treat it algebraically: multiply, divide, and substitute freely.

U-substitution, taught in the CBSE Class 12 Integrals chapter as the method of substitution, is one of the very first integration techniques students learn after the standard formulas, and "integration by substitution class 12 examples" is a heavily searched revision topic. It remains equally essential for solving integral calculus problems in JEE Main and JEE Advanced.