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.
ScanSolve AI Answer
Welcome back سالم — missed you this week. Step 1: Factor the denominator and set up the partial fraction decomposition. First, factor the denominator x^3+4x: x^3+4x = x(x^2+4) Since x^2+4 is an irreducible quadratic factor, the partial fraction decomposition for the integrand is: (x+3)/(x(x^2+4)) = (A)/(x) + (Bx+C)/(x^2+4) Step 2: Solve for the constants A, B, and C. Multiply both sides by x(x^2+4) to clear the denominators: x+3 = A(x^2+4) + (Bx+C)x Expand the right side: x+3 = Ax^2+4A + Bx^2+Cx Group terms by powers of x: x+3 = (A+B)x^2 + Cx + 4A Equate the coefficients of corresponding powers of x on both sides: For x^2: A+B = 0 (1) For x: C = 1 (2) For the constant term: 4A = 3 (3) From equation (3), we find A: A = (3)/(4) From equation (2), we find C: C = 1 Substitute the value of A into equation (1): (3)/(4) + B = 0 B = -(3)/(4) The constants are A = (3)/(4), B = -(3)/(4), and C = 1. These match the values given in the image. Step 3: Rewrite the integral using the partial fractions. Substitute the values of A, B, and C back into the partial fraction decomposition: (x+3)/(x(x^2+4)) = (3/4)/(x) + ((-3/4)x+1)/(x^2+4) This can be split into three simpler fractions: (x+3)/(x(x^2+4)) = (3)/(4x) - (3x)/(4(x^2+4)) + (1)/(x^2+4) Now, we can integrate each term. Step 4: Integrate each term separately. The integral becomes: ( (3)/(4x) - (3x)/(4(x^2+4)) + (1)/(x^2+4) ) dx = (3)/(4x) dx - (3x)/(4(x^2+4)) dx + (1)/(x^2+4) dx Integrate the first term: (3)/(4x) dx = (3)/(4) (1)/(x) dx = (3)/(4) |x| Integrate the second term. Let u = x^2+4, so du = 2x \, dx x \, dx = (1)/(2) du: (3x)/(4(x^2+4)) dx = (3)/(4) (1)/(x^2+4) (x \, dx) = (3)/(4) (1)/(u) ((1)/(2) du) = (3)/(8) (1)/(u) du = (3)/(8) |u| = (3)/(8) (x^2+4) Integrate the third term using the formula (1)/(x^2+a^2) dx = (1)/(a) ((x)/(a)). Here a^2=4, so a=2: (1)/(x^2+4) dx = (1)/(2) ((x)/(2)) Step 5: Combine the results. Combine the results from integrating each term and add the constant of integration C: (x+3)/(x^3+4x) dx = (3)/(4) |x| - (3)/(8) (x^2+4) + (1)/(2) ((x)/(2)) + C The final answer is (3)/(4) |x| - (3)/(8) (x^2+4) + (1)/(2) ((x)/(2)) + C. Drop the next question.