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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You're on a roll — Here are the integrations for each expression: a) Integrate x^2. Step 1: Apply the power rule for integration, x^n \,dx = x^n+1n+1 + C. x^2 \,dx = x^2+12+1 + C = (x^3)/(3) + C The integral is (x^3)/(3) + C. b) Integrate 4x^5. Step 1: Apply the constant multiple rule and the power rule. 4x^5 \,dx = 4 x^5 \,dx = 4 (x^5+15+1) + C = 4 ((x^6)/(6)) + C = (2x^6)/(3) + C The integral is (2x^6)/(3) + C. c) Integrate 2sqrt(x). Step 1: Rewrite the expression using fractional exponents. 2sqrt(x) = 2x^1/2 Step 2: Apply the constant multiple rule and the power rule. 2x^1/2 \,dx = 2 x^1/2 \,dx = 2 (x^1/2+11/2+1) + C = 2 (x^3/23/2) + C = 2 · (2)/(3) x^3/2 + C = (4)/(3) x^3/2 + C The integral is (4)/(3) x^3/2 + C. d) Integrate 7. Step 1: Apply the rule for integrating a constant, k \,dx = kx + C. 7 \,dx = 7x + C The integral is 7x + C. e) Integrate x^3+x^2+x+1. Step 1: Integrate each term separately using the power rule. (x^3+x^2+x+1) \,dx = x^3 \,dx + x^2 \,dx + x \,dx + 1 \,dx = x^3+13+1 + x^2+12+1 + x^1+11+1 + x + C = (x^4)/(4) + (x^3)/(3) + (x^2)/(2) + x + C The integral is (x^4)/(4) + (x^3)/(3) + (x^2)/(2) + x + C. f) Integrate x^3/5. Step 1: Apply the power rule for integration. x^3/5 \,dx = x^3/5+13/5+1 + C = x^8/58/5 + C = (5)/(8) x^8/5 + C The integral is (5)/(8) x^8/5 + C. i) Integrate (x^6)/(2). Step 1: Rewrite the expression and apply the constant multiple rule and the power rule. (x^6)/(2) \,dx = (1)/(2) x^6 \,dx = (1)/(2) (x^6+16+1) + C = (1)/(2) ((x^7)/(7)) + C = (x^7)/(14) + C The integral is (x^7)/(14) + C. j) Integrate x-sqrt(x). Step 1: Rewrite the expression using fractional exponents. x-sqrt(x) = x-x^1/2 Step 2: Integrate each term separately using the power rule. (x-x^1/2) \,dx = x \,dx - x^1/2 \,dx = x^1+11+1 - x^1/2+11/2+1 + C = (x^2)/(2) - x^3/23/2 + C = (x^2)/(2) - (2)/(3) x^3/2 + C The integral is (x^2)/(2) - (2)/(3) x^3/2 + C. k) Integrate ((x+3)^2)/(2x^4). Step 1: Expand the numerator and simplify the expression. (x+3)^2 = x^2 + 6x + 9 ((x+3)^2)/(2x^4) = (x^2 + 6x + 9)/(2x^4) = (x^2)/(2x^4) + (6x)/(2x^4) + (9)/(2x^4) = (1)/(2)x^-2 + 3x^-3 + (9)/(2)x^-4 Step 2: Integrate each term separately using the power rule. ((1)/(2)x^-2 + 3x^-3 + (9)/(2)x^-4) \,dx = (1)/(2) x^-2 \,dx + 3 x^-3 \,dx + (9)/(2) x^-4 \,dx = (1)/(2) (x^-2+1-2+1) + 3 (x^-3+1-3+1) + (9)/(2) (x^-4+1-4+1) + C = (1)/(2) (x^-1-1) + 3 (x^-2-2) + (9)/(2) (x^-3-3) + C = -(1)/(2)x^-1 - (3)/(2)x^-2 - (3)/(2)x^-3 + C Step 3: Rewrite the terms with positive exponents. = -(1)/(2x) - (3)/(2x^2) - (3)/(2x^3) + C The integral is -(1)/(2x) - (3)/(2x^2) - (3)/(2x^3) + C. Got more? Send 'em!