for the antiderivative of the function presented as two separate fractions: (3x2+2x)/(7x2) and (4)/(9x+2). Given the visual separation and the absence of an explicit operator (like $

Step 1: Interpret the function. The problem asks for the antiderivative of the function presented as two separate fractions: (3x^2+2x)/(7x^2) and (4)/(9x+2). Given the visual separation and the absence of an explicit operator (like × or ·), we will assume the function is the sum of these two terms. So, we need to find the integral of f(x) = (3x^2+2x)/(7x^2) + (4)/(9x+2). Step 2: Simplify the first term of the function. (3x^2+2x)/(7x^2) = (x(3x+2))/(7x^2) For x ≠ 0, we can cancel an x from the numerator and denominator: = (3x+2)/(7x) Now, split this into two separate fractions: = (3x)/(7x) + (2)/(7x) = (3)/(7) + (2)/(7) · (1)/(x) Step 3: Rewrite the entire function for integration. f(x) = (3)/(7) + (2)/(7x) + (4)/(9x+2) Step 4: Integrate each term separately. We need to find f(x) \,dx = ( (3)/(7) + (2)/(7x) + (4)/(9x+2) ) \,dx. Integrate the first term: (3)/(7) \,dx = (3)/(7)x Integrate the second term: (2)/(7x) \,dx = (2)/(7) (1)/(x) \,dx = (2)/(7) |x| Integrate the third term: For (4)/(9x+2) \,dx, use a u-substitution. Let u = 9x+2. Then, du = 9 \,dx, which means dx = (1)/(9) \,du. Substitute these into the integral: (4)/(u) ((1)/(9) \,du) = (4)/(9) (1)/(u) \,du = (4)/(9) |u| Substitute back u = 9x+2: = (4)/(9) |9x+2| Step 5: Combine all the integrated terms and add the constant of integration, C. f(x) \,dx = (3)/(7)x + (2)/(7) |x| + (4)/(9) |9x+2| + C The final answer is (3)/(7)x + (2)/(7) |x| + (4)/(9) |9x+2| + C. Last free one today — make it count tomorrow, or type /upgrade for unlimited.