JEE Mathematics 2023 Question with Solution

Given:

Find: Evaluate the integral and identify the correct option.

Factor the denominator:

So,

Use the substitution $u=e^x$, so $du=e^x\,dx$. The limits change as: when $x=0$, $u=1$ and when $x=1$, $u=e$. Hence,

Thus,

Using partial fraction decomposition and simplifying, we get:

Therefore, the value of the integral is $\log_e\left(\frac{32}{27}\right)$. The correct option is A.

Given:

Find: A quick route to the final value.

The useful observation is to rewrite the cubic expression in terms of $e^x$. Once the denominator is factorised, the integral becomes a rational function of $u=e^x$. This works because exponentials of the form $e^{3x}, e^{2x}, e^x$ reduce to powers of a single variable after substitution.

After setting $u=e^x$, the integral changes to a standard rational integral in $u$, and partial fractions lead directly to logarithmic terms. Evaluating between the transformed limits $1$ and $e$ gives

Therefore, the correct option is A.