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 for the first 9 questions. 1) Evaluate (2-sqrt(x))^2 dx Step 1: Expand the integrand. (2-sqrt(x))^2 = 4 - 4sqrt(x) + x = 4 - 4x^1/2 + x Step 2: Integrate each term using the power rule x^n dx = x^n+1n+1 + C. (4 - 4x^1/2 + x) dx = 4x - 4x^3/23/2 + (x^2)/(2) + C Step 3: Simplify the expression. = 4x - (8)/(3)x^3/2 + (1)/(2)x^2 + C The final answer is (1)/(2)x^2 - (8)/(3)x^3/2 + 4x + C. 2) Evaluate (t^6-t^2)/(t^4) dt Step 1: Simplify the integrand by dividing each term in the numerator by t^4. (t^6-t^2)/(t^4) = (t^6)/(t^4) - (t^2)/(t^4) = t^2 - t^-2 Step 2: Integrate each term using the power rule. (t^2 - t^-2) dt = t^2+12+1 - t^-2+1-2+1 + C Step 3: Simplify the expression. = (t^3)/(3) - t^-1-1 + C = (t^3)/(3) + (1)/(t) + C The final answer is (t^3)/(3) + (1)/(t) + C. 3) Evaluate (x^2+1)/(sqrt(x)) dx Step 1: Rewrite the integrand by dividing each term in the numerator by sqrt(x) = x^1/2. (x^2+1)/(sqrt(x)) = (x^2)/(x^1/2) + (1)/(x^1/2) = x^2-1/2 + x^-1/2 = x^3/2 + x^-1/2 Step 2: Integrate each term using the power rule. (x^3/2 + x^-1/2) dx = x^3/2+13/2+1 + x^-1/2+1-1/2+1 + C Step 3: Simplify the expression. = x^5/25/2 + x^1/21/2 + C = (2)/(5)x^5/2 + 2x^1/2 + C The final answer is (2)/(5)x^5/2 + 2sqrt(x) + C. 4) Evaluate 4(sqrt(u) + [3]u) du Step 1: Rewrite the terms with fractional exponents. 4(sqrt(u) + [3]u) = 4(u^1/2 + u^1/3) Step 2: Integrate each term using the power rule. 4(u^1/2 + u^1/3) du = 4 ( u^1/2+11/2+1 + u^1/3+11/3+1 ) + C Step 3: Simplify the expression. = 4 ( u^3/23/2 + u^4/34/3 ) + C = 4 ( (2)/(3)u^3/2 + (3)/(4)u^4/3 ) + C = (8)/(3)u^3/2 + 3u^4/3 + C The final answer is (8)/(3)u^3/2 + 3u^4/3 + C. 5) Evaluate (5)/(sqrt(x)) dx Step 1: Rewrite the integrand with a fractional exponent. (5)/(sqrt(x)) = 5x^-1/2 Step 2: Integrate using the power rule. 5x^-1/2 dx = 5 x^-1/2+1-1/2+1 + C Step 3: Simplify the expression. = 5 x^1/21/2 + C = 10x^1/2 + C = 10sqrt(x) + C The final answer is 10sqrt(x) + C. 6) Evaluate [sqrt(t) - (2)/(sqrt(t))] dt Step 1: Rewrite the terms with fractional exponents. sqrt(t) - (2)/(sqrt(t)) = t^1/2 - 2t^-1/2 Step 2: Integrate each term using the power rule. (t^1/2 - 2t^-1/2) dt = t^1/2+11/2+1 - 2t^-1/2+1-1/2+1 + C Step 3: Simplify the expression. = t^3/23/2 - 2t^1/21/2 + C = (2)/(3)t^3/2 - 4t^1/2 + C The final answer is (2)/(3)t^3/2 - 4sqrt(t) + C. 7) Evaluate (x^4-1)/(x^2+1) dx Step 1: Factor the numerator using the difference of squares formula a^2-b^2 = (a-b)(a+b). x^4-1 = (x^2)^2 - 1^2 = (x^2-1)(x^2+1) Step 2: Simplify the integrand. (x^4-1)/(x^2+1) = ((x^2-1)(x^2+1))/(x^2+1) = x^