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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Step 1: Evaluate (x^2 + 1) \,dx We use the power rule for integration, x^n \,dx = x^n+1n+1 + C. (x^2 + 1) \,dx = x^2 \,dx + 1 \,dx = x^2+12+1 + x + C = (x^3)/(3) + x + C The integral is (x^3)/(3) + x + C. Step 2: Evaluate (dx)/([3]x^2) First, rewrite the integrand using exponent notation: [3]x^2 = x^2/3. Then, (1)/(x^2/3) = x^-2/3. x^-2/3 \,dx Apply the power rule for integration: = x^-2/3 + 1-2/3 + 1 + C = x^1/31/3 + C = 3x^1/3 + C = 3[3]x + C The integral is 3[3]x + C. Step 3: Evaluate (3x^2 + 2x + 5) \,dx Integrate each term using the power rule: (3x^2 + 2x + 5) \,dx = 3x^2 \,dx + 2x \,dx + 5 \,dx = 3x^2+12+1 + 2x^1+11+1 + 5x + C = 3(x^3)/(3) + 2(x^2)/(2) + 5x + C = x^3 + x^2 + 5x + C The integral is x^3 + x^2 + 5x + C. Step 4: Evaluate (x^4 + 3x^3 - 4x)/(x^2) \,dx First, divide each term in the numerator by x^2: (x^4)/(x^2) + (3x^3)/(x^2) - (4x)/(x^2) = x^4-2 + 3x^3-2 - 4x^1-2 = x^2 + 3x - 4x^-1 Now integrate the simplified expression: (x^2 + 3x - 4x^-1) \,dx = x^2 \,dx + 3x \,dx - 4x^-1 \,dx = x^2+12+1 + 3x^1+11+1 - 4|x| + C = (x^3)/(3) + 3(x^2)/(2) - 4|x| + C The integral is (x^3)/(3) + (3x^2)/(2) - 4|x| + C. Step 5: Evaluate x^2 (3x^2 + 4x) \,dx First, expand the integrand: x^2 (3x^2 + 4x) = 3x^2 · x^2 + 4x · x^2 = 3x^4 + 4x^3 Now integrate the expanded expression: (3x^4 + 4x^3) \,dx = 3x^4 \,dx + 4x^3 \,dx = 3x^4+14+1 + 4x^3+13+1 + C = 3(x^5)/(5) + 4(x^4)/(4) + C = (3x^5)/(5) + x^4 + C The integral is (3x^5)/(5) + x^4 + C. Step 6: Evaluate (x^2 - 1)/(x) \,dx First, divide each term in the numerator by x: (x^2)/(x) - (1)/(x) = x - (1)/(x) Now integrate the simplified expression: (x - (1)/(x)) \,dx = x \,dx - (1)/(x) \,dx = x^1+11+1 - |x| + C = (x^2)/(2) - |x| + C The integral is (x^2)/(2) - |x| + C. Step 7: Evaluate (x-2)(2x+3)(x-5) \,dx First, expand the product of the three binomials: Multiply the first two terms: (x-2)(2x+3) = x(2x) + x(3) - 2(2x) - 2(3) = 2x^2 + 3x - 4x - 6 = 2x^2 - x - 6 Now multiply the result by (x-5): (2x^2 - x - 6)(x-5) = 2x^2(x-5) - x(x-5) - 6(x-5) = (2x^3 - 10x^2) - (x^2 - 5x) - (6x - 30) = 2x^3 - 10x^2 - x^2 + 5x - 6x + 30 = 2x^3 - 11x^2 - x + 30 Now integrate the expanded polynomial: (2x^3 - 11x^2 - x + 30) \,dx = 2x^3 \,dx - 11x^2 \,dx - x \,dx + 30 \,dx = 2x^3+13+1 - 11x^2+12+1 - x^1+11+1 + 30x + C = 2(x^4)/(4) - 11(x^3)/(3) - (x^2)/(2) + 30x + C = (x^4)/(2) - (11x^3)/(3) - (x^2)/(2) + 30x + C The integral is (x^4)/(2) - (11x^3)/(3) - (x^2)/(2) + 30x + C.