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
Here are the solutions to the integration problems. 1. Step 1: Apply the power rule for integration, x^n \,dx = x^n+1n+1 + C, and k \,dx = kx + C. (x^7 + 1) \,dx = x^7+17+1 + x + C Step 2: Simplify the expression. = (x^8)/(8) + x + C The student's answer is ✅ Correct. (x^8)/(8) + x + C 2. Step 1: Rewrite the integrand using exponent rules. [3]x^2 = x^2/3, so (1)/([3]x^2) = x^-2/3. (dx)/([3]x^2) = x^-2/3 \,dx Step 2: Apply the power rule for integration, x^n \,dx = x^n+1n+1 + C. Here n = -(2)/(3). = x^-2/3 + 1-2/3 + 1 + C = x^1/31/3 + C Step 3: Simplify the expression. = 3x^1/3 + C The student's answer is ✅ Correct. 3x^1/3 + C 3. Step 1: Apply the power rule for integration to each term. (3x^3 + 2x^2 - x + 5) \,dx = 3x^3+13+1 + 2x^2+12+1 - x^1+11+1 + 5x + C Step 2: Simplify each term. = (3x^4)/(4) + (2x^3)/(3) - (x^2)/(2) + 5x + C The student's answer is ✅ Correct. (3x^4)/(4) + (2x^3)/(3) - (x^2)/(2) + 5x + C 4. Step 1: Simplify the integrand by dividing each term in the numerator by x^2. (x^2 + 3x^3 - 4x)/(x^2) \,dx = ((x^2)/(x^2) + (3x^3)/(x^2) - (4x)/(x^2)) \,dx = (1 + 3x - 4x^-1) \,dx Step 2: Apply the power rule for integration and the rule for (1)/(x) \,dx = |x| + C. = x + 3x^1+11+1 - 4|x| + C Step 3: Simplify the expression. = x + (3x^2)/(2) - 4|x| + C The student's answer is ❌ Incorrect. The first term of the integrand was simplified incorrectly. x + (3x^2)/(2) - 4|x| + C