Question

Which of the following integrals is un-bounded?

• A. \( \int_0^{\frac{\pi}{4}} \tan(x) \, dx \)
• B. \( \int_0^{\infty} e^{-x} \, dx \)
• C. \( \int_0^a \frac{1}{a - x} \, dx \)
• D. \( \int_0^{\infty} x e^x \, dx \)

Hint:

When evaluating integrals, watch out for points where the integrand becomes infinite, as these often lead to unbounded integrals.

Answer 1

The Correct Option is C

The integral \( \int_0^a \frac{1}{a - x} \, dx \) is unbounded because as \( x \) approaches \( a \), the integrand tends to infinity. This creates a singularity at \( x = a \), making the integral unbounded.
For the other options, the integrals are bounded as they do not exhibit any singularities. Hence, the correct answer is option (3).