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Section 5.4 : More Substitution Rule

Evaluate each of the following integrals.

1. \( \displaystyle \int {{3x\cos \left( {4 - {x^2}} \right) - 8x\sqrt {4 - {x^2}} \,dx}}\)
2. \( \displaystyle \int {{\frac{4}{{{{\left( {9 + 6t} \right)}^5}}} + \frac{{13}}{{9 + 6t}}\,dt}}\)
3. \( \displaystyle \int {{\left( {6 - 5w} \right){{\bf{e}}^{12w - 5{w^{\,2}}}} + \left( {20w - 24} \right){{\sec }^2}\left( {12w - 5{w^2}} \right)\,dw}}\)
4. \( \displaystyle \int {{\frac{{\sin \left( {1 + \ln \left( {2x} \right)} \right) - \sqrt {1 + \ln \left( {2x} \right)} }}{x}\,dx}}\)
5. \( \displaystyle \int {{17\left( {x{{\bf{e}}^x} + {{\bf{e}}^x}} \right)\sin \left( {x{{\bf{e}}^x}} \right) - 14\sin \left( x \right)\,dx}}\)
6. \( \displaystyle \int {{\frac{1}{{3t}} + \sec \left( {9t} \right)\tan \left( {9t} \right){{\bf{e}}^{\sec \left( {9t} \right)}}\,dt}}\)
7. \( \displaystyle \int {{8{w^2} + \frac{{\sin \left( w \right) + \cos \left( w \right)}}{{\sin \left( w \right) - \cos \left( w \right)}}\,dw}}\)
8. \( \displaystyle \int {{8 + \left( {3 + {x^6}} \right)\cos \left( {21x + {x^7}} \right) + 9{x^2} - 4\sqrt x \,dx}}\)
9. \( \displaystyle \int {{\sin \left( y \right)\cos \left( y \right)\sqrt {3 + {{\sin }^2}\left( y \right)} + 5{{\bf{e}}^y}\,dy}}\)
10. \( \displaystyle \int {{\sin \left( {2 - t} \right) + 8\cos \left( {5t} \right) - {{\bf{e}}^{3t}}\,dt}}\)
11. \( \displaystyle \int {{\frac{{4{x^2} - 1}}{{\sqrt[4]{{6x - 8{x^3}}}}} + 9x{{\bf{e}}^{{x^{\,2}}}}\,dx}}\)
12. \( \displaystyle \int {{{z^3} + \sqrt {4 - 3z} - 4\sec \left( {8z} \right)\tan \left( {8z} \right)\,dz}}\)
13. \( \displaystyle \int {{\frac{{17}}{{6 - w}} + \sin \left( w \right)\sin \left[ {1 + \cos \left( w \right)} \right]\,dw}}\)
14. \( \displaystyle \int {{\frac{{\sqrt {1 + 2\ln \left( {7x} \right)} }}{x} + \frac{{10{x^3}}}{{{x^4} + 9}}\,dx}}\)
15. \( \displaystyle \int {{x\sin \left( {{x^2}} \right)\left[ {{{\cos }^4}\left( {{x^2}} \right) + 8{{\cos }^2}\left( {{x^2}} \right) - 10} \right]\,dx}}\)
16. \( \displaystyle \int {{\csc \left( {\frac{t}{2}} \right)\cot \left( {\frac{t}{2}} \right)\left[ {{{\csc }^6}\left( {\frac{t}{2}} \right) + 3{{\csc }^4}\left( {\frac{t}{2}} \right) - 8\csc \left( {\frac{t}{2}} \right)} \right]\,dt}}\)
17. \( \displaystyle \int {{\frac{{3 + 7y}}{{{y^2} + 3}}\,dy}}\)
18. \( \displaystyle \int {{\frac{{15z + 27}}{{100{z^2} + 11}}\,dz}}\)
19. \( \displaystyle \int {{\frac{{8x + 1}}{{\sqrt {16 - {x^2}} }}\,dx}}\)
20. \( \displaystyle \int {{\frac{{2 - w}}{{\sqrt {25 - 2{w^2}} }}\,dw}}\)
21. \( \displaystyle \int {{\frac{{9{z^5}}}{{2 + 3{z^3}}}\,dz}}\)
22. \( \displaystyle \int {{4{t^{15}}\sqrt {1 - {t^8}} \,dt}}\)
23. \( \displaystyle \int {{\cot \left( x \right)\,dx}}\)
24. \( \displaystyle \int {{\csc \left( x \right)\,dx}}\)
25. \( \displaystyle \int {{\frac{x}{{1 + {x^4}}}\,dx}}\)
26. \( \displaystyle \int {{{{\bf{e}}^{8t}}{{\left( {4 + {{\bf{e}}^{4t}}} \right)}^{ - 3}}\,dt}}\)
27. \( \displaystyle \int {{{x^8}\sqrt {2 - {x^3}} \,dx}}\)