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 solutions to the integration problems: 1. Integrate ((1)/(2)x)\,dx Step 1: Use a substitution. Let u = (1)/(2)x. Then, du = (1)/(2)\,dx, which means dx = 2\,du. Step 2: Substitute u and dx into the integral. (u) (2\,du) = 2 (u)\,du Step 3: Integrate with respect to u. 2 (-(u)) + C = -2(u) + C Step 4: Substitute back u = (1)/(2)x. -2((1)/(2)x) + C 2. Integrate x\,dx Step 1: Rewrite x in terms of sine and cosine. ( x)/( x)\,dx Step 2: Use a substitution. Let u = x. Then, du = x\,dx. Step 3: Substitute u and du into the integral. (1)/(u)\,du Step 4: Integrate with respect to u. |u| + C Step 5: Substitute back u = x. | x| + C 3. Integrate x sqrt(1 + x^2)\,dx Step 1: Use a substitution. Let u = 1 + x^2. Then, du = 2x\,dx, which means x\,dx = (1)/(2)\,du. Step 2: Substitute u and x\,dx into the integral. sqrt(u) ((1)/(2)\,du) = (1)/(2) u^1/2\,du Step 3: Integrate with respect to u. (1)/(2) (u^1/2 + 11/2 + 1) + C = (1)/(2) (u^3/23/2) + C = (1)/(2) · (2)/(3) u^3/2 + C = (1)/(3) u^3/2 + C Step 4: Substitute back u = 1 + x^2. (1)/(3)(1 + x^2)^3/2 + C 4. Integrate e^x (3 + e^x)^4\,dx Step 1: Use a substitution. Let u = 3 + e^x. Then, du = e^x\,dx. Step 2: Substitute u and du into the integral. u^4\,du Step 3: Integrate with respect to u. u^4+14+1 + C = (u^5)/(5) + C Step 4: Substitute back u = 3 + e^x. (1)/(5)(3 + e^x)^5 + C 5. Integrate (1)/(1 + x^2)\,dx Step 1: Recognize this as a standard integral. The derivative of x is (1)/(1 + x^2). Step 2: Apply the integration formula directly. x + C 6. Integrate ^2 x\,dx Step 1: Use the trigonometric identity ^2 x = (1 + (2x))/(2). (1 + (2x))/(2)\,dx = (1)/(2) (1 + (2x))\,dx Step 2: Integrate term by term. (1)/(2) ( 1\,dx + (2x)\,dx) Step 3: Integrate each term. For (2x)\,dx, let u = 2x, so du = 2\,dx, or dx = (1)/(2)\,du. (2x)\,dx = (u) ((1)/(2)\,du) = (1)/(2) (u) + C_1 = (1)/(2) (2x) + C_1 And 1\,dx = x + C_2. Step 4: Combine the results. (1)/(2) (x + (1)/(2)(2x)) + C (1)/(2)x + (1)/(4)(2x) + C 7. Integrate ^3 x\,dx Step 1: Rewrite ^3 x using the identity ^2 x = 1 - ^2 x. ^2 x · x\,dx = (1 - ^2 x) x\,dx Step 2: Use a substitution. Let u = x. Then, du = x\,dx. Step 3: Substitute u and du into the integral. (1 - u^2)\,du Step 4: Integrate with respect to u. u - (u^3)/(3) + C Step 5: Substitute back u = x. x - (1)/(3)^3 x + C What's next?