Evaluate the following integrals: 1. integral (2x+3)/(x2-3x+2) dx 2a. integral (x7 + 4x6 - 2x5 + (1)/(x) + e^x + x) dx

Welcome back Kheny’s — missed you this week. Here are the solutions to the integration problems: 1. Evaluate (2x+3)/(x^2-3x+2) dx Step 1: Factor the denominator and decompose the integrand into partial fractions. The denominator x^2-3x+2 factors as (x-1)(x-2). We set up the partial fraction decomposition: (2x+3)/((x-1)(x-2)) = (A)/(x-1) + (B)/(x-2) Multiplying both sides by (x-1)(x-2) gives: 2x+3 = A(x-2) + B(x-1) To find A, set x=1: 2(1)+3 = A(1-2) + B(1-1) 5 = -A A = -5 To find B, set x=2: 2(2)+3 = A(2-2) + B(2-1) 7 = B So, the integral becomes: ( (-5)/(x-1) + (7)/(x-2) ) dx Step 2: Integrate each term. (-5)/(x-1) dx = -5 |x-1| (7)/(x-2) dx = 7 |x-2| Step 3: Combine the results and add the constant of integration. (2x+3)/(x^2-3x+2) dx = -5 |x-1| + 7 |x-2| + C The final answer is -5 |x-1| + 7 |x-2| + C. 2. Evaluate the following: a) (x^7 + 4x^6 - 2x^5 + (1)/(x) + e^x + x) dx Step 1: Integrate each term using the power rule, logarithm rule, exponential rule, and trigonometric rule for integration. x^n dx = x^n+1n+1 + C (1)/(x) dx = |x| + C e^x dx = e^x + C x dx = - x + C Step 2: Apply the integration rules to each term. x^7 dx = x^7+17+1 = (x^8)/(8) 4x^6 dx = 4 x^6+16+1 = (4x^7)/(7) -2x^5 dx = -2 x^5+15+1 = -2 (x^6)/(6) = -(x^6)/(3) (1)/(x) dx = |x| e^x dx = e^x x dx = - x Step 3: Combine all the integrated terms and add the constant of integration. (x^7 + 4x^6 - 2x^5 + (1)/(x) + e^x + x) dx = (x^8)/(8) + (4x^7)/(7) - (x^6)/(3) + |x| + e^x - x + C The final answer is (x^8)/(8) + (4x^7)/(7) - (x^6)/(3) + |x| + e^x - x + C. b) (1)/(x^2+49) dx Step 1: Recognize the form of the integral. This integral matches the standard form (1)/(x^2+a^2) dx = (1)/(a) ((x)/(a)) + C. Step 2: Identify the value of a. In this case, a^2 = 49, so a = 7. Step 3: Apply the formula. (1)/(x^2+49) dx = (1)/(7) ((x)/(7)) + C The final answer is (1)/(7) ((x)/(7)) + C. Drop the next question!