NIMCET 2021 — Mathematics PYQ

1. Substitution

Let's set the entire "top" tower as our variable $u$:

2. Differentiating both sides

Recall that the derivative of $a^f(x)$ is $a^f(x)\cdot \ln (a)\cdot f^{\prime }(x)$. Applying the chain rule repeatedly:

3. Simplify the expression

Combining the $\ln (3)$ terms, we get:

Now, notice that the expression in the parentheses is exactly what we have in our original integral! So, we can rewrite it as:

4. Substitute back into the Integral

The integral now becomes very simple:

5. Integrate and Final Result

Since $\frac{1}{(\ln 3{)}^3}$ is just a constant:

Substituting $u=3^{3^{3^x}}$ back:

Final Answer