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Section 5-7 : Computing Definite Integrals

  1. Evaluate each of the following integrals.
    1. \( \displaystyle \int{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\)
    2. \( \displaystyle \int_{{ - 3}}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\)
    3. \( \displaystyle \int_{1}^{4}{{\cos \left( x \right) - \frac{3}{{{x^5}}}\,dx}}\)
    Solution

Evaluate each of the following integrals, if possible. If it is not possible clearly explain why it is not possible to evaluate the integral.

  1. \( \displaystyle \int_{1}^{6}{{12{x^3} - 9{x^2} + 2\,dx}}\) Solution
  2. \( \displaystyle \int_{{ - 2}}^{1}{{5{z^2} - 7z + 3\,dz}}\) Solution
  3. \( \displaystyle \int_{3}^{0}{{15{w^4} - 13{w^2} + w\,dw}}\) Solution
  4. \( \displaystyle \int_{1}^{4}{{\frac{8}{{\sqrt t }} - 12\sqrt {{t^3}} \,dt}}\) Solution
  5. \( \displaystyle \int_{1}^{2}{{\frac{1}{{7z}} + \frac{{\sqrt[3]{{{z^2}}}}}{4} - \frac{1}{{2{z^3}}}\,dz}}\) Solution
  6. \( \displaystyle \int_{{ - 2}}^{4}{{{x^6} - {x^4} + \frac{1}{{{x^2}}}\,dx}}\) Solution
  7. \( \displaystyle \int_{{ - 4}}^{{ - 1}}{{{x^2}\left( {3 - 4x} \right)\,dx}}\) Solution
  8. \( \displaystyle \int_{2}^{1}{{\frac{{2{y^3} - 6{y^2}}}{{{y^2}}}\,dy}}\) Solution
  9. \( \displaystyle \int_{0}^{{\frac{\pi }{2}}}{{7\sin \left( t \right) - 2\cos \left( t \right)\,dt}}\) Solution
  10. \( \displaystyle \int_{0}^{\pi }{{\sec \left( z \right)\tan \left( z \right) - 1\,dz}}\) Solution
  11. \( \displaystyle \int_{{\frac{\pi }{6}}}^{{\frac{\pi }{3}}}{{2{{\sec }^2}\left( w \right) - 8\csc \left( w \right)\cot \left( w \right)\,dw}}\) Solution
  12. \( \displaystyle \int_{0}^{2}{{{{\bf{e}}^x} + \frac{1}{{{x^2} + 1}}\,dx}}\) Solution
  13. \( \displaystyle \int_{{ - 5}}^{{ - 2}}{{7{{\bf{e}}^y} + \frac{2}{y}\,dy}}\) Solution
  14. \( \displaystyle \int_{0}^{4}{{f\left( t \right)\,dt}}\) where \(f\left( t \right) = \left\{ {\begin{array}{*{20}{c}}{2t}&{t > 1}\\{1 - 3{t^2}}&{t \le 1}\end{array}} \right.\) Solution
  15. \( \displaystyle \int_{{ - 6}}^{1}{{g\left( z \right)\,dz}}\) where \(g\left( z \right) = \left\{ {\begin{array}{*{20}{c}}{2 - z}&{z > - 2}\\{4{{\bf{e}}^z}}&{z \le - 2}\end{array}} \right.\) Solution
  16. \( \displaystyle \int_{3}^{6}{{\left| {2x - 10} \right|\,dx}}\) Solution
  17. \( \displaystyle \int_{{ - 1}}^{0}{{\left| {4w + 3} \right|\,dw}}\) Solution