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Prove some properties of the improper integrals ∫ (sin x) / x and ∫ (cos t) / t2

by RoRi
  1. Prove that the following improper integral converges:

        \[ \int_{0^+}^1 \frac{\sin x}{x} \, dx. \]

  2. Prove that

        \[ \lim_{x \to 0^+} x \int_x^1 \frac{\cos t}{t^2} \, dt = 1. \]

  3. Determine the convergence or divergence of the improper integral

        \[ \int_{0^+}^1 \frac{\cos t}{t^2} \, dt. \]

Incomplete.

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