![Solve the following integrals: a) integral [7x3 + 2x + 1] \,dx b) integral [7x3 + 2x] \,dx c) integral [5x2 + x] \,dx](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1779200233018-68f2e317715e8c18.png&w=3840&q=75)
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
Hey Ambe, good to see you again. Here are the solutions to the integrals: a) For the integral [7x^3 + 2x + 1] \,dx: Step 1: Apply the power rule for integration, x^n \,dx = x^n+1n+1 + C, and the constant rule, k \,dx = kx + C. 7x^3 \,dx = 7 x^3+13+1 = 7 (x^4)/(4) 2x \,dx = 2 x^1+11+1 = 2 (x^2)/(2) = x^2 1 \,dx = x Step 2: Combine the results and add the constant of integration, C. [7x^3 + 2x + 1] \,dx = (7)/(4)x^4 + x^2 + x + C The final answer is (7)/(4)x^4 + x^2 + x + C. b) For the integral [7x^3 + 2x] \,dx: Step 1: Apply the power rule for integration to each term. 7x^3 \,dx = 7 x^3+13+1 = 7 (x^4)/(4) 2x \,dx = 2 x^1+11+1 = 2 (x^2)/(2) = x^2 Step 2: Combine the results and add the constant of integration, C. [7x^3 + 2x] \,dx = (7)/(4)x^4 + x^2 + C The final answer is (7)/(4)x^4 + x^2 + C. c) For the integral [5x^2 + x] \,dx: Step 1: Apply the power rule for integration to each term. 5x^2 \,dx = 5 x^2+12+1 = 5 (x^3)/(3) x \,dx = x^1+11+1 = (x^2)/(2) Step 2: Combine the results and add the constant of integration, C. [5x^2 + x] \,dx = (5)/(3)x^3 + (1)/(2)x^2 + C The final answer is (5)/(3)x^3 + (1)/(2)x^2 + C. d) For the integral [8x^3 + 2] \,dx: Step 1: Apply the power rule and constant rule for integration to each term. 8x^3 \,dx = 8 x^3+13+1 = 8 (x^4)/(4) = 2x^4 2 \,dx = 2x Step 2: Combine the results and add the constant of integration, C. [8x^3 + 2] \,dx = 2x^4 + 2x + C The final answer is 2x^4 + 2x + C. Send me the next one 📸