Show that the area under the curve is infinite show that

Question

1. Show that _-∞∫^∞ (1 + x) / (1 + x²) dx diverges, but lim_t→∞-t∫^t (1 + x) / (1 + x²) dx = π.

2. Use the Comparison Theorem to decide if these integrals converge or diverge.

a. ₁∫^∞ (4 + e^-x) / x dx
b. ₀∫^∞ arctan x / (2 + e^x) dx

3. The Gamma function is defined by Γ(t) = ₀∫^∞ x^t-1 e^-x dx. Use integration by parts to prove that Γ(z + 1) = z Γ(z).

4. Consider the curve y = 1/x for x ≥ 1.

a. Show that the area under the curve is infinite.

b. Show that the volume of the solid of revolution enclosed by the curve is finite, and compute it. [This solid of revolution is known as Gabriel's Horn.]