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
Okay, Felix, let's integrate these expressions. a) To evaluate (1)/(x) dx: Step 1: Recall the standard integral for (1)/(x). The integral of (1)/(x) with respect to x is the natural logarithm of the absolute value of x. (1)/(x) dx = |x| + C The integral is |x| + C. b) To evaluate (3/x + 7x^2 + 7e^x - x) dx: Step 1: Integrate each term separately using the sum/difference rule for integrals. ((3)/(x) + 7x^2 + 7e^x - x) dx = (3)/(x) dx + 7x^2 dx + 7e^x dx - x dx Step 2: Apply the power rule for integration ( x^n dx = x^n+1n+1), the integral of e^x, and the integral of x. (3)/(x) dx = 3 |x| 7x^2 dx = 7 x^2+12+1 = 7 (x^3)/(3) 7e^x dx = 7e^x x dx = x Step 3: Combine the results and add the constant of integration, C. (3/x + 7x^2 + 7e^x - x) dx = 3 |x| + (7)/(3)x^3 + 7e^x - x + C The integral is 3 |x| + (7)/(3)x^3 + 7e^x - x + C. c) To evaluate (2x-3)^2 dx: Step 1: Expand the term (2x-3)^2. (2x-3)^2 = (2x)^2 - 2(2x)(3) + (-3)^2 = 4x^2 - 12x + 9 Step 2: Integrate the expanded polynomial term by term. (4x^2 - 12x + 9) dx = 4x^2 dx - 12x dx + 9 dx Step 3: Apply the power rule for integration to each term. 4x^2 dx = 4 x^2+12+1 = 4 (x^3)/(3) -12x dx = -12 x^1+11+1 = -12 (x^2)/(2) = -6x^2 9 dx = 9x Step 4: Combine the results and add the constant of integration, C. (2x-3)^2 dx = (4)/(3)x^3 - 6x^2 + 9x + C The integral is (4)/(3)x^3 - 6x^2 + 9x + C. 3 done, 2 left today. You're making progress.