This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems. b) Integrate: (1)/(x^2sqrt(x^2-9)) dx Step 1: Use trigonometric substitution. The integral contains sqrt(x^2 - a^2) where a=3. Let x = 3 . Then dx = 3 \, d. And sqrt(x^2 - 9) = sqrt((3 )^2 - 9) = sqrt(9 ^2 - 9) = sqrt(9(^2 - 1)) = sqrt(9 ^2 ) = 3 . Step 2: Substitute these into the integral and simplify. (1)/((3 )^2 (3 )) (3 \, d) = (3 )/(9 ^2 · 3 ) \, d = (3 )/(27 ^2 ) \, d = (1)/(9 ) \, d = (1)/(9) \, d Step 3: Integrate with respect to . = (1)/(9) + C Step 4: Convert back to x. From x = 3 , we have = (x)/(3). This means = (3)/(x). Consider a right triangle where the hypotenuse is x and the adjacent side is 3. The opposite side is sqrt(x^2 - 3^2) = sqrt(x^2 - 9). Therefore, = oppositehypotenuse = sqrt(x^2 - 9)x. Substitute this back into the result from Step 3. (1)/(9) sqrt(x^2 - 9)x + C The final answer is sqrt(x^2 - 9)9x + C. 10. a) For any three sets A, B and C prove that: (A B) - C = (A - C) (B - C). To prove this set equality, we will show that each side is a subset of the other. Step 1: Prove (A B) - C (A - C) (B - C). Let x (A B) - C. By the definition of set difference, this means x (A B) and x C. By the definition of union, x A or x B. So, we have (x A or x B) and x C. Using the distributive property of "and" over "or", this is equivalent to: (x A and x C) or (x B and x C). By the definition of set difference, this means (x A - C) or (x B - C). By the definition of union, this means x (A - C) (B - C). Therefore, (A B) - C (A - C) (B - C). Step 2: Prove (A - C) (B - C) (A B) - C. Let x (A - C) (B - C). By the definition of union, this means x (A - C) or x (B - C). By the definition of set difference, this means (x A and x C) or (x B and x C). Using the distributive property of "and" over "or", this is equivalent to: (x A or x B) and x C. By the definition of union, this means x (A B). So, we have x (A B) and x C. By the definition of set difference, this means x (A B) - C. Therefore, (A - C) (B - C) (A B) - C. Step 3: Conclude the proof. Since we have shown that (A B) - C (A - C) (B - C) and (A - C) (B - C) (A B) - C, the two sets are equal. Thus, (A B) - C = (A - C) (B - C) is proven. 3 done, 2 left today. You're making progress.