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√x²+25 tan(8) = Video Example 4 EXAMPLE 3 Find 1 / x² √ x ² + 25 ° dx. SOLUTION Let x = 5 tan(0), -π/2 < 0</2. Then dx = Thus we have x²+25= √√25 tan² (8) + 25 = √25 sec (0) = 5 sec(0)| = 5 sec(0). dx x²x² + 25 sec(8) tan²(8) = 1. - 21/5). tan²(0) To evaluate this trigonometric integral we put everything in terms of sin(0) and cos(0). dx x²√x² + 25 1 dx x²√x² + 25 25 tan² (8) 5 sec(8) = sin²(8) Therefore, making the substitution u = sin(8), we have cos² (8) sin²(0) 25 sin²(8) du (-)+c C de 25 sin(8) -csc(8) + + C + C. de. We use the figure to determine that csc(8)=√√x² + 25/x and so + C. de and de