Section 5.7 : Computing Definite Integrals

1. Evaluate each of the following integrals.
   1. \( \displaystyle \int{{3{z^2} - 4 + \frac{4}{{{z^2}}}\,dz}}\)
   2. \( \displaystyle \int_{1}^{4}{{3{z^2} - 4 + \frac{4}{{{z^2}}}\,dz}}\)
   3. \( \displaystyle \int_{{ - 2}}^{1}{{3{z^2} - 4 + \frac{4}{{{z^2}}}\,dz}}\)
Evaluate each of the following integrals.
2. 1. \( \displaystyle \int{{6x + \frac{1}{{3x}}\,dx}}\)
   2. \( \displaystyle \int_{0}^{7}{{6x + \frac{1}{{3x}}\,dx}}\)
   3. \( \displaystyle \int_{3}^{7}{{6x + \frac{1}{{3x}}\,dx}}\)
3. Evaluate each of the following integrals.
   1. \( \displaystyle \int{{\sin \left( y \right) + {{\sec }^2}\left( y \right)\,dy}}\)
   2. \( \displaystyle \int_{0}^{{\frac{\pi }{4}}}{{\sin \left( y \right) + {{\sec }^2}\left( y \right)\,dy}}\)
   3. \( \displaystyle \int_{0}^{{\frac{{2\pi }}{3}}}{{\sin \left( y \right) + {{\sec }^2}\left( y \right)\,dy}}\)

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. \( \displaystyle \int_{0}^{3}{{10t - 6{t^2} + 9\,dt}}\)
2. \( \displaystyle \int_{{ - 1}}^{4}{{24{z^2} + 5{z^4}dz}}\)
3. \( \displaystyle \int_{1}^{0}{{9w - 3{w^2} + 4{w^3}\,dw}}\)
4. \( \displaystyle \int_{{ - 3}}^{{ - 1}}{{15{t^2} - 10t - 2dt}}\)
5. \( \displaystyle \int_{{ - 2}}^{4}{{{v^3} - 7{v^2} + 3v\,dv}}\)
6. \( \displaystyle \int_{0}^{{16}}{{9\sqrt x + 10\,\sqrt[4]{x}\,dx}}\)
7. \( \displaystyle \int_{{ - 1}}^{2}{{8\,\,\sqrt[3]{z} - 12\,\,\sqrt[5]{z}\,dz}}\)
8. \( \displaystyle \int_{1}^{4}{{\sqrt {{y^5}} - \frac{1}{{\sqrt[3]{y}}}\,dy}}\)
9. \( \displaystyle \int_{1}^{4}{{\frac{6}{{{x^3}}} - \frac{1}{{3{x^2}}}\,dx}}\)
10. \( \displaystyle \int_{6}^{{ - 3}}{{8{w^3} - 25{w^4} + \frac{4}{{3{w^5}}}\,dw}}\)
11. \( \displaystyle \int_{{ - 1}}^{{ - 3}}{{\frac{4}{{3{z^2}}} - \frac{6}{{{z^3}}}\,dz}}\)
12. \( \displaystyle \int_{0}^{6}{{\left( {3 - t} \right)\left( {2{t^2} + 3} \right)\,dt}}\)
13. \( \displaystyle \int_{4}^{1}{{\sqrt x \left( {x - 2{x^2} + 1} \right)\,dx}}\)
14. \( \displaystyle \int_{2}^{5}{{\frac{{6{z^5} - 8{z^4} + 2{z^2}}}{{{z^4}}}\,dz}}\)
15. \( \displaystyle \int_{{ - 2}}^{{ - 4}}{{\frac{{9{x^4} - 8{x^3} + x}}{{3{x^2}}}\,dx}}\)
16. \( \displaystyle \int_{{ - 8}}^{2}{{\frac{{7{v^{10}} + 4{v^6} - 3{v^2}}}{{{v^5}}}\,dv}}\)
17. \( \displaystyle \int_{1}^{2}{{\frac{{\left( {y - 2} \right)\left( {y + 2} \right)}}{{{y^2}}}\,dy}}\)
18. \( \displaystyle \int_{0}^{{\frac{\pi }{4}}}{{8{{\sec }^2}\left( t \right) + 2\sec \left( t \right)\tan \left( t \right)\,dt}}\)
19. \( \displaystyle \int_{{ - \,\frac{\pi }{3}}}^{{\frac{\pi }{6}}}{{3\cos \left( w \right) + \sin \left( w \right)\,dw}}\)
20. \( \displaystyle \int_{{ - \,\frac{\pi }{4}}}^{{\frac{\pi }{4}}}{{12{{\sec }^2}\left( y \right) - 9{{\csc }^2}\left( y \right)\,dy}}\)
21. \( \displaystyle \int_{{\frac{{2\pi }}{3}}}^{{\frac{\pi }{4}}}{{3\sin \left( v \right) + 8\csc \left( v \right)\cot \left( v \right)\,dv}}\)
22. \( \displaystyle \int_{{ - 3}}^{1}{{4x - 7{{\bf{e}}^x}\,dx}}\)
23. \( \displaystyle \int_{{ - 2}}^{1}{{\frac{{4{{\bf{e}}^{2w}} + 4w\,{{\bf{e}}^w}}}{{{{\bf{e}}^w}}}\,dw}}\)
24. \( \displaystyle \int_{0}^{{\frac{1}{2}}}{{\frac{3}{{\sqrt {1 - {x^2}} }} + \frac{7}{{{x^2} + 1}}\,dx}}\)
25. \( \displaystyle \int_{{ - 2}}^{3}{{5\sin \left( t \right) + \frac{1}{{\sqrt {1 - {t^2}} }}\,dt}}\)
26. \( \displaystyle \int_{6}^{{10}}{{\frac{4}{z} + \frac{1}{{2{z^2}}}\,dz}}\)
27. \( \displaystyle \int_{1}^{6}{{2{x^3} + \frac{3}{{8x}}\,dx}}\)
28. \( \displaystyle \int_{{ - 4}}^{{ - 1}}{{f\left( t \right)\,dt}}\) where \(f\left( t \right) = \left\{ {\begin{array}{*{20}{c}}{9 + 6{t^2}}&{t > - 3}\\{8t}&{t \le - 3}\end{array}} \right.\)
29. \( \displaystyle \int_{{ - 2}}^{4}{{g\left( x \right)\,dx}}\) where \(g\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{9 - 2{{\bf{e}}^x}}&{x > 0}\\{8\sin \left( x \right)}&{x \le 0}\end{array}} \right.\)
30. \( \displaystyle \int_{4}^{9}{{h\left( w \right)\,dw}}\) where \(h\left( w \right) = \left\{ {\begin{array}{*{20}{c}}4&{w > 6}\\{3w + 1}&{w \le 6}\end{array}} \right.\)
31. \( \displaystyle \int_{{ - 1}}^{7}{{f\left( x \right)\,dx}}\) where \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{9{x^2}} & {x > 5}\\{ - 7} & {1 < x \le 5}\\{3 - 8x} & {x \le 1}\end{array}} \right.\)
32. \( \displaystyle \int_{{ - 3}}^{1}{{\left| {8 + 4x} \right|\,dx}}\)
33. \( \displaystyle \int_{2}^{8}{{\left| {3v - 12} \right|\,dv}}\)
34. \( \displaystyle \int_{0}^{6}{{\left| {10 - 2z} \right|\,dz}}\)
35. \( \displaystyle \int_{{ - 3}}^{6}{{\left| {{t^2} - 4} \right|\,dt}}\)