Techniques of Integration - Integration by the Reverse Chain Rule

1. \(\displaystyle \displaystyle \displaystyle\int{{{x}^{2}}{{({{x}^{3}}+3)}^{3}}}\,\mathrm{d}x\)
2. \(\displaystyle \displaystyle \displaystyle\int{(5{{x}^{4}}+4x)({{x}^{5}}+2{{x}^{2}}}+1{{)}^{3}}\,\mathrm{d}x\)
3. \(\displaystyle \displaystyle \displaystyle\int{\frac{3x-2}{{{\left( 3{{x}^{2}}-4x-1 \right)}^{3}}}}\,\mathrm{d}x\)
4. \(\displaystyle \displaystyle \displaystyle\int{\frac{{{x}^{2}}}{\sqrt{{{x}^{3}}+1}}\,\mathrm{d}x}\)
5. \(\displaystyle \displaystyle \displaystyle\int{\frac{21{{x}^{2}}+14}{\sqrt{{{x}^{3}}+2x+1}}\,\mathrm{d}x}\)
6. \(\displaystyle \displaystyle \displaystyle\int{\frac{x}{{{\sqrt{1-2{{x}^{2}}}}^{5}}}\,\mathrm{d}x}\)
7. \(\displaystyle \displaystyle \displaystyle\int{{{\left( \frac{x}{\sqrt{3{{x}^{4}}-1}} \right)}^{3}}\,\mathrm{d}x}\)
8. \(\displaystyle \displaystyle \displaystyle\int{3{{x}^{3}}{{(2{{x}^{4}}-5)}^{8}}}\,\mathrm{d}x\)
9. \(\displaystyle \displaystyle \displaystyle\int{{{e}^{3x}}{{({{e}^{3x}}+5)}^{11}}}\,\mathrm{d}x\)
10. \(\displaystyle \displaystyle \displaystyle\int{\frac{{{x}^{3}}}{{{x}^{4}}+3}\text{}\hspace{0.5em}}\mathrm{d}x\)

11. \(\displaystyle \displaystyle \displaystyle\int{\frac{2x+1}{\left( 2{{x}^{2}}+2x+5 \right)}}\,\mathrm{d}x\)
12. \(\displaystyle \displaystyle \displaystyle\int{\frac{3x}{1+{{x}^{2}}}}\,\mathrm{d}x\)
13. \(\displaystyle \displaystyle \displaystyle\int{\frac{2x+1}{{{x}^{2}}+x+1}}\,\mathrm{d}x\)
14. \(\displaystyle \displaystyle \displaystyle\int{\frac{1}{\sqrt{x}{{(\sqrt{x}+1)}^{9}}}}\,\mathrm{d}x\)
15. \(\displaystyle \displaystyle \displaystyle\int{\frac{1}{\sqrt{x}{{e}^{\sqrt{x}}}}}\,\mathrm{d}x\)
16. \(\displaystyle \displaystyle \displaystyle\int{\frac{{{\left( \ln x \right)}^{8}}}{x}}\,\mathrm{d}x\)
17. \(\displaystyle \displaystyle \displaystyle\int{\frac{1}{x\ln {{x}^{3}}}}\,\mathrm{d}x\)
18. \(\displaystyle \displaystyle \displaystyle\int{x}\left( {{e}^{{{x}^{2}}+3}} \right)\,\mathrm{d}x\)
19. \(\displaystyle \displaystyle \displaystyle\int{x{{e}^{{{x}^{2}}}}}\left( {{e}^{{{x}^{2}}-1}} \right)\,\mathrm{d}x\)
20. \(\displaystyle \displaystyle \displaystyle\int{\frac{1}{1+{{e}^{-x}}}}\,\mathrm{d}x\)

21. \(\displaystyle \displaystyle \displaystyle\int{{{x}^{7}}\cos ({{x}^{8}}+2)}\,\mathrm{d}x\)
22. \(\displaystyle \displaystyle \displaystyle\int{\tan 3x}\,\mathrm{d}x\)
23. \(\displaystyle \displaystyle \displaystyle\int{\cot x}\,\mathrm{d}x\)
24. \(\displaystyle \displaystyle \displaystyle\int{\frac{{{\tan }^{5}}x}{{{\cos }^{2}}x}}\,\mathrm{d}x\)
25. \(\displaystyle \displaystyle \displaystyle\int{\frac{1}{{{\sin }^{2}}x\tan x}}\,\mathrm{d}x\)
26. \(\displaystyle \displaystyle \displaystyle\int{\sec x\tan x}\,\mathrm{d}x\)
27. \(\displaystyle \displaystyle \displaystyle\int{\text{cosec}\hspace{0.5em}x\cot x}\,\mathrm{d}x\)
28. \(\displaystyle \displaystyle \displaystyle\int{\frac{\sin x\cos x}{{{\sin }^{2}}x+7{{\cos }^{2}}x}}\,\mathrm{d}x\)
29. \(\displaystyle \displaystyle \displaystyle\int{\frac{\sin 2x}{\sqrt{5{{\sin }^{2}}x+4{{\cos }^{2}}x}}}\,\mathrm{d}x\)
30. \(\displaystyle \displaystyle \displaystyle\int{(\sin x){{e}^{\cos x}}}\,\mathrm{d}x\)
31. \(\displaystyle \displaystyle \displaystyle\int{x{{e}^{{{x}^{2}}}}}\,\mathrm{d}x\)
32. \(\displaystyle \displaystyle \displaystyle\int{{{2}^{x}}}\,\mathrm{d}x\)
33. \(\displaystyle \displaystyle \displaystyle\int{{{5}^{x}}}\,\mathrm{d}x\)