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Section 7.2 : Integrals Involving Trig Functions

Evaluate each of the following integrals.

1. \( \displaystyle \int{{{{\sin }^3}\left( {\frac{2}{3}x} \right){{\cos }^4}\left( {\frac{2}{3}x} \right)\,dx}}\) Solution
2. \( \displaystyle \int{{{{\sin }^8}\left( {3z} \right){{\cos }^5}\left( {3z} \right)\,dz}}\) Solution
3. \( \displaystyle \int{{{{\cos }^4}\left( {2t} \right)\,dt}}\) Solution
4. \( \displaystyle \int_{\pi }^{{2\pi }}{{{{\cos }^3}\left( {\frac{1}{2}w} \right){{\sin }^5}\left( {\frac{1}{2}w} \right)\,dw}}\) Solution
5. \( \displaystyle \int{{{{\sec }^6}\left( {3y} \right){{\tan }^2}\left( {3y} \right)\,dy}}\) Solution
6. \( \displaystyle \int{{{{\tan }^3}\left( {6x} \right){{\sec }^{10}}\left( {6x} \right)\,dx}}\) Solution
7. \( \displaystyle \int_{0}^{{\frac{\pi }{4}}}{{{{\tan }^7}\left( z \right){{\sec }^3}\left( z \right)\,dz}}\) Solution
8. \( \displaystyle \int{{\cos \left( {3t} \right)\sin \left( {8t} \right)\,dt}}\) Solution
9. \( \displaystyle \int_{1}^{3}{{\sin \left( {8x} \right)\sin \left( x \right)\,dx}}\) Solution
10. \( \displaystyle \int{{\cot \left( {10z} \right){{\csc }^4}\left( {10z} \right)\,dz}}\) Solution
11. \( \displaystyle \int{{{{\csc }^6}\left( {\frac{1}{4}w} \right){{\cot }^4}\left( {\frac{1}{4}w} \right)\,dw}}\) Solution
12. \( \displaystyle \int{{\frac{{{{\sec }^4}\left( {2t} \right)}}{{{{\tan }^9}\left( {2t} \right)}}\,dt}}\) Solution
13. \( \displaystyle \int{{\frac{{2 + 7{{\sin }^3}\left( z \right)}}{{{{\cos }^2}\left( z \right)}}\,dz}}\) Solution
14. \( \displaystyle \int{{\left[ {9{{\sin }^5}\left( {3x} \right) - 2{{\cos }^3}\left( {3x} \right)} \right]{{\csc }^4}\left( {3x} \right)\,dx}}\) Solution