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 for the first two questions: Question 1: We need to evaluate the integral: (dx)/(sqrt(9+x^2)) Step 1: Identify the appropriate trigonometric substitution. The integral is of the form (du)/(sqrt(a^2+u^2)). Here, a^2=9 a=3 and u^2=x^2 u=x. Let x = 3 . Then, dx = 3 ^2 \, d. Step 2: Substitute into the integral. First, simplify the term under the square root: sqrt(9+x^2) = sqrt(9+(3 )^2) = sqrt(9+9 ^2 ) = sqrt(9(1+^2 )) = sqrt(9 ^2 ) = 3 Now substitute dx and sqrt(9+x^2) into the integral: (3 ^2 \, d)/(3 ) = \, d Step 3: Evaluate the integral. The integral of is a standard integral: \, d = | + | + C Step 4: Convert back to the original variable x. From our substitution x = 3 , we have = (x)/(3). To find , we can construct a right triangle where the opposite side is x and the adjacent side is 3. The hypotenuse is sqrt(x^2+3^2) = sqrt(x^2+9). Thus, = hypotenuseadjacent = sqrt(x^2+9)3. Substitute these expressions back into the result from Step 3: |sqrt(x^2+9)3 + (x)/(3)| + C = |x+sqrt(x^2+9)3| + C Using the logarithm property ((A)/(B)) = A - B: = |x+sqrt(x^2+9)| - 3 + C Since - 3 is a constant, we can absorb it into the arbitrary constant C. = |x+sqrt(x^2+9)| + C The final answer is |x+sqrt(x^2+9)| + C. --- Question 2: We need to evaluate the integral: (3\,dx)/(sqrt(1+9x^2)) Step 1: Identify the appropriate trigonometric substitution. The integral is of the form (k\,du)/(sqrt(a^2+u^2)). Here, a^2=1 a=1. For 9x^2, let u^2 = 9x^2 u = 3x. Let 3x = 1 = . Then x = (1)/(3) . And dx = (1)/(3) ^2 \, d. Step 2: Substitute into the integral. First, simplify the term under the square root: sqrt(1+9x^2) = sqrt(1+( )^2) = sqrt(1+^2 ) = sqrt(^2 ) = Now substitute dx and sqrt(1+9x^2) into the integral: (3 (1)/(3) ^2 \, d) = (^2 \, d)/( ) = \, d Step 3: Evaluate the integral. The integral of is a standard integral: \, d = | + | + C Step 4: Convert back to the original variable x. From our substitution 3x = , we have = 3x. To find , we can construct a right triangle where the opposite side is 3x and the adjacent side is 1. The hypotenuse is sqrt((3x)^2+1^2) = sqrt(9x^2+1). Thus, = hypotenuseadjacent = sqrt(9x^2+1)1 = sqrt(9x^2+1). Substitute these expressions back into the result from Step 3: |sqrt(9x^2+1) + 3x| + C The final answer is |3x+sqrt(9x^2+1)| + C.