JEE Mathematics 2023 Question with Solution

Given:

Find: $l + m + n$ when the integral is written in the required form.

Use the substitution

Then

So the integral becomes

because when $x = 0$, $t = 0$ and when $x = 1$, $t = 11$.

Now integrate:

Comparing with the required form, the solution gives

Hence

Therefore, the required answer is $63$.

Given: the integrand contains a polynomial factor multiplied by a power of another expression.

Find: how to convert it quickly into a direct substitution form.

Notice that

So the polynomial factor is exactly proportional to the derivative of the inner bracket. This is why the substitution works immediately.

Therefore,

Thus the compared values are $l = 48$, $m = 8$, $n = 7$, and hence $l + m + n = 63$.