3 Integrate the following expressions using Table 13.1: \( \begin{array}{ll}\text { (a) } \sin 4 x & \text { (b) } \sin 9 t \\ \text { (c) } \sin \left(\frac{x}{2}\right) & \text { (d) } \sin \left(\frac{2 t}{5}\right) \\ \text { (c) } \cos 7 x & \text { (f) } \cos (-3 x) \\ \text { (g) } \cos \left(\frac{5 t}{3}\right) & \text { (h) } \tan 9 x \\ \text { (i) } \operatorname{cosec} 2 x & \text { (j) } \sec 5 t \\ \text { (k) } \cot 8 y & \text { (l) } \cos (5 t+1)\end{array} \)

To integrate the given expressions, we will use standard integration formulas for trigonometric functions. Here are the integrals for each expression: ### (a) \( \sin 4x \) \[ \int \sin 4x \, dx = -\frac{1}{4} \cos 4x + C \] ### (b) \( \sin 9t \) \[ \int \sin 9t \, dt = -\frac{1}{9} \cos 9t + C \] ### (c) \( \sin \left(\frac{x}{2}\right) \) \[ \int \sin \left(\frac{x}{2}\right) \, dx = -2 \cos \left(\frac{x}{2}\right) + C \] ### (d) \( \sin \left(\frac{2t}{5}\right) \) \[ \int \sin \left(\frac{2t}{5}\right) \, dt = -\frac{5}{2} \cos \left(\frac{2t}{5}\right) + C \] ### (e) \( \cos 7x \) \[ \int \cos 7x \, dx = \frac{1}{7} \sin 7x + C \] ### (f) \( \cos (-3x) \) Using the property \( \cos(-\theta) = \cos(\theta) \): \[ \int \cos (-3x) \, dx = \int \cos 3x \, dx = \frac{1}{3} \sin 3x + C \] ### (g) \( \cos \left(\frac{5t}{3}\right) \) \[ \int \cos \left(\frac{5t}{3}\right) \, dt = \frac{3}{5} \sin \left(\frac{5t}{3}\right) + C \] ### (h) \( \tan 9x \) \[ \int \tan 9x \, dx = -\frac{1}{9} \ln |\cos 9x| + C \] ### (i) \( \operatorname{cosec} 2x \) \[ \int \operatorname{cosec} 2x \, dx = -\frac{1}{2} \ln |\operatorname{cosec} 2x + \cot 2x| + C \] ### (j) \( \sec 5t \) \[ \int \sec 5t \, dt = \frac{1}{5} \ln |\sec 5t + \tan 5t| + C \] ### (k) \( \cot 8y \) \[ \int \cot 8y \, dy = \frac{1}{8} \ln |\sin 8y| + C \] ### (l) \( \cos (5t + 1) \) \[ \int \cos (5t + 1) \, dt = \frac{1}{5} \sin (5t + 1) + C \] ### Summary of Integrals 1. \( \int \sin 4x \, dx = -\frac{1}{4} \cos 4x + C \) 2. \( \int \sin 9t \, dt = -\frac{1}{9} \cos 9t + C \) 3. \( \int \sin \left(\frac{x}{2}\right) \, dx = -2 \cos \left(\frac{x}{2}\right) + C \) 4. \( \int \sin \left(\frac{2t}{5}\right) \, dt = -\frac{5}{2} \cos \left(\frac{2t}{5}\right) + C \) 5. \( \int \cos 7x \, dx = \frac{1}{7} \sin 7x + C \) 6. \( \int \cos (-3x) \, dx = \frac{1}{3} \sin 3x + C \) 7. \( \int \cos \left(\frac{5t}{3}\right) \, dt = \frac{3}{5} \sin \left(\frac{5t}{3}\right) + C \) 8. \( \int \tan 9x \, dx = -\frac{1}{9} \ln |\cos 9x| + C \) 9. \( \int \operatorname{c