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Section 5-8 : Substitution Rule for Definite Integrals

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. \( \displaystyle \int_{0}^{1}{{3\left( {4x + {x^4}} \right){{\left( {10{x^2} + {x^5} - 2} \right)}^6}\,dx}}\) Solution
  2. \( \displaystyle \int_{0}^{{\frac{\pi }{4}}}{{\frac{{8\cos \left( {2t} \right)}}{{\sqrt {9 - 5\sin \left( {2t} \right)} }}\,dt}}\) Solution
  3. \( \displaystyle \int_{\pi }^{0}{{\sin \left( z \right){{\cos }^3}\left( z \right)\,dz}}\) Solution
  4. \( \displaystyle \int_{1}^{4}{{\sqrt w \,{{\bf{e}}^{1 - \sqrt {{w^{\,3}}} }}\,dw}}\) Solution
  5. \( \displaystyle \int_{{ - 4}}^{{ - 1}}{{\sqrt[3]{{5 - 2y}} + \frac{7}{{5 - 2y}}\,dy}}\) Solution
  6. \( \displaystyle \int_{{ - 1}}^{2}{{{x^3} + {{\bf{e}}^{\frac{1}{4}x}}\,dx}}\) Solution
  7. \( \displaystyle \int_{\pi }^{{\frac{{3\pi }}{2}}}{{6\sin \left( {2w} \right) - 7\cos \left( w \right)dw}}\) Solution
  8. \( \displaystyle \int_{1}^{5}{{\frac{{2{x^3} + x}}{{{x^4} + {x^2} + 1}} - \frac{x}{{{x^2} - 4}}\,dx}}\) Solution
  9. \( \displaystyle \int_{{ - 2}}^{0}{{t\sqrt {3 + {t^2}} + \frac{3}{{{{\left( {6t - 1} \right)}^2}}}\,dt}}\) Solution
  10. \( \displaystyle \int_{{ - 2}}^{1}{{{{\left( {2 - z} \right)}^3} + \sin \left( {\pi z} \right){{\left[ {3 + 2\cos \left( {\pi z} \right)} \right]}^3}\,dz}}\) Solution