JAMIA 2026 — Mathematics PYQ

Step 1: Choose the substitution

Let us substitute the nested function:

$$t=\log (\log x)$$

Step 2: Differentiate with respect to $x$

Using the chain rule, differentiate $t$:

$$\frac{dt}{dx}=\frac{1}{\log x}\cdot \frac{d}{dx}(\log x)$$

$$\frac{dt}{dx}=\frac{1}{\log x}\cdot \frac{1}{x}$$

Now, separate the variables to match our integral:

$$dt=\frac{dx}{x\log x}$$

Step 3: Substitute back into the integral

Notice how perfectly the remaining part of our original integral matches our expression for $dt$:

$$I=\int \frac{1}{\log (\log x)}\cdot \left(\frac{dx}{x\log x}\right)$$

Replacing $\log (\log x)$ with $t$ and $\frac{dx}{x\log x}$ with $dt$:

$$I=\int \frac{1}{t}dt$$

Step 4: Integrate and substitute back

The standard integral of $\frac{1}{t}$ is $\log t$:

$$I=\log (t)+C$$

(where $C$ is the constant of integration)

Finally, replace $t$ back with its original value $\log (\log x)$:

$$I=\log (\log (\log x))+C$$

Correct Answer:

(d) $\log (\log (\log x))$