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Section 5.3 : Substitution Rule for Indefinite Integrals

For problems 1 – 31 evaluate the given integral.

1. \( \displaystyle \int {{12v{{\left( {7 + 6{v^2}} \right)}^9}\,dv}}\)
2. \( \displaystyle \int {{\left( {4{x^3} - 12x} \right){{\left( {{x^4} - 6{x^2}} \right)}^{ - 3}}\,dx}}\)
3. \( \displaystyle \int {{\left( {{z^2} - 4} \right){{\left( {12z - {z^3}} \right)}^4}\,dz}}\)
4. \( \displaystyle \int {{7{z^2}{{\left( {14 + 8{z^3}} \right)}^{ - 5}}\,dz}}\)
5. \( \displaystyle \int {{3\left( {{y^6} - 4{y^{ - 3}}} \right){{\left( {{y^7} + 14{y^{ - 2}} - 7} \right)}^6}\,dy}}\)
6. \( \displaystyle \int {{\left( {\frac{1}{2}{x^3} - 1} \right)\sqrt {8x - {x^4}} \,dx}}\)
7. \( \displaystyle \int {{\left( {6{w^{ - 4}} + 12{w^{ - 7}}} \right)\,\,\sqrt[4]{{{w^{ - 3}} + {w^{ - 6}}}}\,dw}}\)
8. \( \displaystyle \int {{\cos \left( {7t} \right)\,dt}}\)
9. \( \displaystyle \int {{\left( {v - 2{v^3}} \right)\cos \left( {{v^2} - {v^4}} \right)\,dv}}\)
10. \( \displaystyle \int {{\sqrt z \sin \left( {1 + \sqrt {{z^3}} } \right)\,dz}}\)
11. \( \displaystyle \int {{{{\csc }^2}\left( {1 + 2x} \right)\,dx}}\)
12. \( \displaystyle \int {{7{w^{ - 5}}\sec \left( {{w^{ - 4}}} \right)\tan \left( {{w^{ - 4}}} \right)\,dw}}\)
13. \( \displaystyle \int {{\left( {2 - {t^2}} \right){{\bf{e}}^{6t - {t^{\,3}}}}\,dt}}\)
14. \( \displaystyle \int {{12{z^{ - 2}}{{\bf{e}}^{4 + {z^{ - 1}}}}\,dz}}\)
15. \( \displaystyle \int {{\frac{1}{{4 - 9w}}\,dw}}\)
16. \( \displaystyle \int {{\frac{{9y}}{{{y^2} + 3}}\,dy}}\)
17. \( \displaystyle \int {{\frac{{6{x^2} - 10{x^4}}}{{{x^5} - {x^3}}}\,dx}}\)
18. \( \displaystyle \int {{\frac{1}{t}\sin \left( {1 - \ln \left( t \right)} \right)\,dt}}\)
19. \( \displaystyle \int {{\left[ {6v - 18\sin \left( {6v} \right)} \right]\,\,\,\sqrt[5]{{{v^2} + \cos \left( {6v} \right)}}\,dv}}\)
20. \( \displaystyle \int {{{{\bf{e}}^{ - 3z}}\sec \left( {{{\bf{e}}^{ - 3z}}} \right)\tan \left( {{{\bf{e}}^{ - 3z}}} \right)\,dz}}\)
21. \( \displaystyle \int {{\left( {\cos \left( x \right) + \sin \left( x \right)} \right){{\bf{e}}^{\sin \left( x \right) - \cos \left( x \right)}}\,dx}}\)
22. \( \displaystyle \int {{\frac{{{{\left[ {\ln \left( {{w^2}} \right)} \right]}^4}}}{w}\,dw}}\)
23. \( \displaystyle \int {{\cos \left( v \right)\cos \left( {1 + \sin \left( v \right)} \right)\,dv}}\)
24. \( \displaystyle \int {{\frac{{y + \sin \left( {2y} \right)}}{{{y^2} - \cos \left( {2y} \right)}}\,dy}}\)
25. \( \displaystyle \int {{{{\sec }^7}\left( t \right)\tan \left( t \right)\,dt}}\)
26. \( \displaystyle \int {{{{\bf{e}}^z}{{\sec }^2}\left( {{{\bf{e}}^z}} \right){{\left[ {1 + \tan \left( {{{\bf{e}}^z}} \right)} \right]}^{ - 3}}\,dz}}\)
27. \( \displaystyle \int {{\frac{7}{{1 + 5{x^2}}}\,dx}}\)
28. \( \displaystyle \int {{\frac{2}{{3 + 4{t^2}}}\,dt}}\)
29. \( \displaystyle \int {{\frac{1}{{\sqrt {16 - {y^2}} }}\,dy}}\)
30. \( \displaystyle \int {{\frac{3}{{\sqrt {7 - 4{v^2}} }}\,dv}}\)
31. \( \displaystyle \int {{\frac{x}{{1 + {x^4}}}\,dx}}\)
32. Evaluate each of the following integrals.
    1. \( \displaystyle \int {{\frac{1}{{3 + x}}\,dx}}\)
    2. \( \displaystyle \int {{\frac{x}{{3 + {x^2}}}\,dx}}\)
    3. \( \displaystyle \int {{\frac{x}{{{{\left( {3 + {x^2}} \right)}^7}}}\,dx}}\)
    4. \( \displaystyle \int {{\frac{1}{{3 + {x^2}}}\,dx}}\)
33. Evaluate each of the following integrals.
    1. \( \displaystyle \int {{\frac{{4w}}{{25 + 9{w^2}}}\,dw}}\)
    2. \( \displaystyle \int {{\frac{{4w}}{{{{\left( {25 + 9{w^2}} \right)}^3}}}\,dw}}\)
    3. \( \displaystyle \int {{\frac{4}{{25 + 9{w^2}}}\,dw}}\)