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Section 7.1 : Integration by Parts

Evaluate each of the following integrals.

1. \( \displaystyle \int{{8t{{\bf{e}}^{7t}}\,dt}}\)
2. \( \displaystyle \int_{\pi }^{{2\pi }}{{\left( {1 - 3x} \right)\sin \left( {\frac{1}{2}x} \right)\,dx}}\)
3. \( \displaystyle \int_{{ - 1}}^{2}{{{w^2}{{\bf{e}}^{4w}}\,dw}}\)
4. \( \displaystyle \int_{1}^{3}{{{{\left( {2 - x} \right)}^2}\ln \left( {4x} \right)\,dx}}\)
5. \( \displaystyle \int{{\left( {6 + 3z} \right)\cos \left( {1 + 4z} \right)\,dz}}\)
6. \( \displaystyle \int{{2{y^2}\cos \left( {9y} \right)\,dy}}\)
7. \( \displaystyle \int{{\left( {3z + {z^2}} \right)\sin \left( z \right)\,dz}}\)
8. \( \displaystyle \int{{\sqrt {{x^3}} \ln \left( {\sqrt[3]{x}} \right)\,dx}}\)
9. \( \displaystyle \int{{\left( {2{w^2} - w} \right){{\bf{e}}^{7w - 1}}\,dw}}\)
10. \( \displaystyle \int{{9t{{\sec }^2}\left( {2t} \right)\,dt}}\)
11. \( \displaystyle \int_{0}^{{\frac{\pi }{8}}}{{{{\bf{e}}^{ - x}}\sin \left( {4x} \right)\,dx}}\)
12. \( \displaystyle \int{{8{{\tan }^{ - 1}}\left( {2y} \right)\,dy}}\)
13. \( \displaystyle \int{{{{\bf{e}}^{6t}}\cos \left( {2t} \right)\,dt}}\)
14. \( \displaystyle \int{{ - 3{{\sin }^{ - 1}}\left( {10x} \right)\,dx}}\)
15. \( \displaystyle \int{{{{\bf{e}}^{3 - z}}\sin \left( {2 + z} \right)\,dz}}\)
16. \( \displaystyle \int_{{ - 1}}^{0}{{2{x^{17}}{{\bf{e}}^{1 + {x^{\,9}}}}\,dx}}\)
17. \( \displaystyle \int{{9{t^{11}}\cos \left( {1 - {t^6}} \right)\,dt}}\)
18. \( \displaystyle \int{{\frac{{{x^7}}}{{\sqrt {{x^4} + 1} }}\,dx}}\)
19. \( \displaystyle \int{{\left( {5 + {x^4}} \right)\sin \left( {\frac{1}{2}x} \right)\,dx}}\)
20. \( \displaystyle \int{{2{z^5}{{\bf{e}}^{1 - z}}\,dz}}\)
21. \( \displaystyle \int{{\left( {5 + 2{w^3} - {w^5}} \right)\cos \left( {3w} \right)\,dw}}\)