12. Techniques of Integration

For each x > 0, let G(x) = ∫(from 0 to x) e^(-xt) dt. Prove that xG(x) = 1 for each x > 0.

Which of the improper integrals in Exercises 63–68 converge and which diverge?

∫ from 6 to ∞ of (1 / √(θ² + 1)) dθ

Which of the improper integrals in Exercises 63–68 converge and which diverge?

∫ from −∞ to ∞ of (2 / (e^x + e^(−x))) dx