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Section 1-9 : Comparison Test for Improper Integrals

Use the Comparison Test to determine if the following integrals converge or diverge.

  1. \( \displaystyle \int_{4}^{\infty }{{\frac{1}{{\sqrt[5]{{z - 2}}}}\,dz}}\)
  2. \( \displaystyle \int_{0}^{\infty }{{\frac{w}{{\sqrt {{w^6} + 2} }}\,dw}}\)
  3. \( \displaystyle \int_{2}^{\infty }{{\frac{1}{{{{\left( {2w + 3} \right)}^4}}}\,dw}}\)
  4. \( \displaystyle \int_{{12}}^{\infty }{{\frac{{{y^2} - 4y + 2}}{{y - 7}}\,dy}}\)
  5. \( \displaystyle \int_{2}^{\infty }{{\frac{1}{{\ln \left( x \right)}}\,dx}}\) Hint : Sketch the graph of \(y = x\) and \(y = \ln \left( x \right)\) on the same axis system.
  6. \( \displaystyle \int_{2}^{\infty }{{\frac{{\sqrt z - 4{{\sin }^2}\left( z \right)}}{{{z^3}}}\,dz}}\)
  7. \( \displaystyle \int_{{20}}^{\infty }{{\frac{{\sqrt[3]{{2x}} + {{\sin }^2}\left( x \right)}}{{\sqrt x - {{\cos }^2}\left( x \right)}}\,dx}}\)
  8. \( \displaystyle \int_{0}^{\infty }{{\frac{{z{{\bf{e}}^{ - z}}}}{{{z^3} + 1}}\,dz}}\)