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Section 1-8 : Improper Integrals
Determine if each of the following integrals converge or diverge. If the integral converges determine its value.
- \( \displaystyle \int_{4}^{\infty }{{2 - 4x + 6{x^2}\,dx}}\)
- \( \displaystyle \int_{0}^{5}{{\frac{1}{{4w - 20}}\,dw}}\)
- \( \displaystyle \int_{{ - 1}}^{2}{{\frac{3}{{\sqrt[6]{{4 - 2z}}}}\,dz}}\)
- \( \displaystyle \int_{{ - \infty }}^{0}{{x\,{{\bf{e}}^{2 + 3x}}\,dx}}\)
- \( \displaystyle \int_{0}^{\infty }{{x\,{{\bf{e}}^{2 + 3x}}\,dx}}\)
- \( \displaystyle \int_{2}^{\infty }{{\frac{1}{{{x^2} + 1}}\,dx}}\)
- \( \displaystyle \int_{0}^{3}{{\frac{1}{{{z^2} - 4z}}\,dz}}\)
- \( \displaystyle \int_{{ - \infty }}^{1}{{\frac{x}{{{x^2} + 1}}\,dx}}\)
- \( \displaystyle \int_{{ - 1}}^{2}{{\frac{1}{{{y^2} - 2y - 3}}\,dy}}\)
- \( \displaystyle \int_{{ - \infty }}^{0}{{\cos \left( w \right)\,dw}}\)
- \( \displaystyle \int_{{10}}^{\infty }{{\frac{1}{{{{\left( {5 - 2z} \right)}^2}}}\,dz}}\)
- \( \displaystyle \int_{{ - \infty }}^{\infty }{{\frac{{{z^3}}}{{{z^4} + 1}}\,dz}}\)
- \( \displaystyle \int_{1}^{4}{{\frac{1}{{2y - 6}}\,dy}}\)
- \( \displaystyle \int_{1}^{5}{{\frac{1}{{\sqrt[3]{{w - 2}}}}\,dw}}\)
- \( \displaystyle \int_{{ - 2}}^{1}{{\frac{{{{\bf{e}}^{\frac{1}{x}}}}}{{{x^2}}}\,dx}}\)
- \( \displaystyle \int_{{ - \infty }}^{\infty }{{{x^2}{{\bf{e}}^{{x^{\,3}}}}\,dx}}\)
- \( \displaystyle \int_{{ - \infty }}^{\infty }{{\frac{y}{{{{\left( {{y^2} + 1} \right)}^3}}}\,dy}}\)
- \( \displaystyle \int_{0}^{3}{{\frac{{{w^3}}}{{\sqrt {9 - {w^2}} }}\,dw}}\)
- \( \displaystyle \int_{{ - 3}}^{1}{{\frac{1}{{{w^2} + 2w}}\,dw}}\)
- \( \displaystyle \int_{0}^{\infty }{{\frac{{{{\bf{e}}^{\frac{1}{x}}}}}{{{x^2}}}\,dx}}\)
- \( \displaystyle \int_{0}^{\infty }{{\frac{1}{{z{{\left[ {\ln \left( z \right)} \right]}^2}}}\,dz}}\)
- \( \displaystyle \int_{0}^{\infty }{{\frac{1}{{w - 1}}\,dw}}\)