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 evaluations of the given integrals: i) Evaluate (3x+2)\,dx Step 1: Use substitution. Let u = 3x+2. Differentiate u with respect to x: (du)/(dx) = 3 du = 3\,dx dx = (1)/(3)\,du Step 2: Substitute u and dx into the integral. (u) ((1)/(3))\,du = (1)/(3) (u)\,du Step 3: Integrate with respect to u. The integral of (u) is (u). = (1)/(3) (u) + C Step 4: Substitute back u = 3x+2. = (1)/(3) (3x+2) + C The final answer is (1)/(3) (3x+2) + C. ii) Evaluate x^3 x\,dx Step 1: Use integration by parts, u\,dv = uv - v\,du. Choose u = x and dv = x^3\,dx. Then, find du and v: du = (1)/(x)\,dx v = x^3\,dx = (x^4)/(4) Step 2: Apply the integration by parts formula. x^3 x\,dx = ( x)((x^4)/(4)) - ((x^4)/(4))((1)/(x))\,dx = (x^4)/(4) x - (x^3)/(4)\,dx Step 3: Evaluate the remaining integral. = (x^4)/(4) x - (1)/(4) x^3\,dx = (x^4)/(4) x - (1)/(4) ((x^4)/(4)) + C = (x^4)/(4) x - (x^4)/(16) + C Step 4: Factor out common terms. = (x^4)/(16) (4 x - 1) + C The final answer is (x^4)/(4) x - (x^4)/(16) + C. iii) Evaluate (6x)/(x^2+3)\,dx Step 1: Use substitution. Let u = x^2+3. Differentiate u with respect to x: (du)/(dx) = 2x du = 2x\,dx Step 2: Rewrite the integral in terms of u. Notice that 6x\,dx = 3 · (2x\,dx). So, 6x\,dx = 3\,du. The integral becomes: (3\,du)/(u) = 3 (1)/(u)\,du Step 3: Integrate with respect to u. The integral of (1)/(u) is |u|. = 3 |u| + C Step 4: Substitute back u = x^2+3. Since x^2+3 is always positive, the absolute value is not strictly necessary. = 3 (x^2+3) + C The final answer is 3 (x^2+3) + C. That's 2 down. 3 left today — send the next one.