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Section 7.2 : Integrals Involving Trig Functions

Evaluate each of the following integrals.

1. \( \displaystyle \int{{{{\cos }^5}\left( {2t} \right){{\sin }^2}\left( {2t} \right)\,dt}}\)
2. \( \displaystyle \int{{{{\cos }^3}\left( {12x} \right)\,dx}}\)
3. \( \displaystyle \int{{{{\cos }^2}\left( z \right){{\sin }^4}\left( z \right)\,dz}}\)
4. \( \displaystyle \int_{{\frac{\pi }{3}}}^{\pi }{{{{\sin }^5}\left( {\frac{3}{4}w} \right){{\cos }^6}\left( {\frac{3}{4}w} \right)\,dw}}\)
5. \( \displaystyle \int_{0}^{\pi }{{{{\cos }^{11}}\left( {5z} \right){{\sin }^3}\left( {5z} \right)\,dz}}\)
6. \( \displaystyle \int{{{{\sin }^2}\left( {7x} \right)\,dx}}\)
7. \( \displaystyle \int_{0}^{{\frac{\pi }{6}}}{{{{\tan }^3}\left( {8x} \right){{\sec }^3}\left( {8x} \right)\,dx}}\)
8. \( \displaystyle \int{{{{\sec }^8}\left( {\frac{1}{2}t} \right){{\tan }^5}\left( {\frac{1}{2}t} \right)\,dt}}\)
9. \( \displaystyle \int{{{{\sec }^2}\left( {9z} \right){{\tan }^3}\left( {9z} \right)\,dz}}\)
10. \( \displaystyle \int_{{\frac{{3\pi }}{4}}}^{\pi }{{{{\sec }^6}\left( {10t} \right){{\tan }^4}\left( {10t} \right)\,dt}}\)
11. \( \displaystyle \int{{{{\tan }^{12}}\left( {2w} \right){{\sec }^6}\left( {2w} \right)\,dw}}\)
12. \( \displaystyle \int{{{{\cot }^2}\left( {3x} \right){{\csc }^6}\left( {3x} \right)\,dx}}\)
13. \( \displaystyle \int_{{\frac{\pi }{3}}}^{{\frac{{2\pi }}{3}}}{{{{\csc }^3}\left( {\frac{1}{4}w} \right){{\cot }^3}\left( {\frac{1}{4}w} \right)\,dw}}\)
14. \( \displaystyle \int{{{{\csc }^4}\left( {6w} \right)\,dw}}\)
15. \( \displaystyle \int{{{{\csc }^{12}}\left( x \right){{\cot }^5}\left( x \right)\,dx}}\)
16. \( \displaystyle \int{{\cot \left( x \right)\,dx}}\)
17. \( \displaystyle \int{{{{\cot }^3}\left( x \right)\,dx}}\)
18. \( \displaystyle \int{{\csc \left( x \right)\,dx}}\)
19. \( \displaystyle \int{{{{\csc }^3}\left( x \right)\,dx}}\)
20. \( \displaystyle \int_{{ - 2}}^{4}{{\sin \left( {8x} \right)\cos \left( {15x} \right)\,dx}}\)
21. \( \displaystyle \int{{\cos \left( {2x} \right)\cos \left( {24x} \right)\,dx}}\)
22. \( \displaystyle \int{{\sin \left( {13z} \right)\sin \left( {9z} \right)\,dz}}\)
23. \( \displaystyle \int{{\frac{{{{\cos }^5}\left( {2t} \right)}}{{{{\sin }^3}\left( {2t} \right)}}\,dt}}\)
24. \( \displaystyle \int{{\frac{{{{\sin }^3}\left( {2 - x} \right)}}{{{{\cos }^2}\left( {2 - x} \right)}}\,dx}}\)
25. \( \displaystyle \int{{\frac{{{{\sec }^6}\left( {\frac{1}{2}z} \right)}}{{{{\tan }^8}\left( {\frac{1}{2}z} \right)}}\,dz}}\)
26. \( \displaystyle \int{{\frac{{{{\tan }^5}\left( x \right)}}{{{{\sec }^2}\left( x \right)}}\,dx}}\)
27. \( \displaystyle \int{{\frac{{1 + 9{{\cos }^5}\left( {8w} \right)}}{{{{\sin }^2}\left( {8w} \right)}}\,dw}}\)
28. \( \displaystyle \int{{\left( {3 + 7{{\cos }^3}\left( x \right)} \right){{\sin }^2}\left( x \right)\,dx}}\)
29. \( \displaystyle \int{{{{\sin }^3}\left( {9y} \right){{\sec }^2}\left( {9y} \right)\,dy}}\)
30. \( \displaystyle \int{{{{\tan }^5}\left( z \right){{\cos }^5}\left( z \right)\,dz}}\)
31. \( \displaystyle \int{{\left[ {\tan \left( {2t} \right) - {{\sin }^3}\left( {2t} \right)} \right]{{\sec }^3}\left( {2t} \right)\,dt}}\)