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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Welcome back Kheny’s — been a while! Let's pick up where you left off. Here are the evaluations of the integrals: 1) Evaluate (2x)\,dx. Step 1: Use u-substitution. Let u = 2x. Then, differentiate u with respect to x: (du)/(dx) = 2 du = 2\,dx dx = (1)/(2)\,du Step 2: Substitute u and dx into the integral. (u) ((1)/(2)\,du) = (1)/(2) (u)\,du Step 3: Integrate with respect to u. The integral of (u) is (u). (1)/(2) (u) + C Step 4: Substitute back u = 2x. (1)/(2) (2x) + C 2) Evaluate e^2x-4\,dx. Step 1: Use u-substitution. Let u = 2x-4. Then, differentiate u with respect to x: (du)/(dx) = 2 du = 2\,dx dx = (1)/(2)\,du Step 2: Substitute u and dx into the integral. e^u ((1)/(2)\,du) = (1)/(2) e^u\,du Step 3: Integrate with respect to u. The integral of e^u is e^u. (1)/(2) e^u + C Step 4: Substitute back u = 2x-4. (1)/(2) e^2x-4 + C 3) Evaluate e^3x\,dx. Step 1: Use u-substitution. Let u = 3x. Then, 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. e^u ((1)/(3)\,du) = (1)/(3) e^u\,du Step 3: Integrate with respect to u. The integral of e^u is e^u. (1)/(3) e^u + C Step 4: Substitute back u = 3x. (1)/(3) e^3x + C 4) Evaluate (2x+3)^1/2\,dx. Step 1: Use u-substitution. Let u = 2x+3. Then, differentiate u with respect to x: (du)/(dx) = 2 du = 2\,dx dx = (1)/(2)\,du Step 2: Substitute u and dx into the integral. u^1/2 ((1)/(2)\,du) = (1)/(2) u^1/2\,du Step 3: Integrate with respect to u using the power rule x^n\,dx = x^n+1n+1 + C. (1)/(2) · u^1/2+11/2+1 + C = (1)/(2) · u^3/23/2 + C Step 4: Simplify the expression. (1)/(2) · (2)/(3) u^3/2 + C = (1)/(3) u^3/2 + C Step 5: Substitute back u = 2x+3. (1)/(3) (2x+3)^3/2 + C Drop the next question!