Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade None
Question:
If $I$ is the greatest of the definite integrals $I_1 = \int_0^1 e^{-x}\cos^2 x dx, I_2 = \int_0^1 e^{-x^2}\cos^2 x dx, I_3 = \int_0^1 e^{-x^2} dx, I_4 = \int_0^1 e^{-x^2/2} dx$ then:
I = I1
I = I2
I = I3
I = I4
Step-by-Step Solution
Key Concept: Converting to a substitution that creates a standard arctangent form reduces complexity.
Rewrite the numerator as $x - 1 = (x+1)^2 - (x^2 + x + 1)$ to split the integral. Substitute $t^2 = x + \frac{1}{x} + 1$, which gives $2tdt = (1 - \frac{1}{x^2})dx$. The integral simplifies to $2\int \frac{tdt}{(t^2+1)t} = 2\int \frac{dt}{t^2+1}$, yielding $2\tan^{-1}(x + \frac{1}{x} + 1) + C$.
Correct Answer: 3