![Evaluate the integral: integral (x2 dx)/([5]3x3 + 7)](/_next/image?url=https%3A%2F%2Fohijasnjvprzodcmxltt.supabase.co%2Fstorage%2Fv1%2Fobject%2Fpublic%2Fwhatsapp-images%2F1775187514805-b70243798b36028a.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
Step 1: Identify the substitution. Let u = 3x^3 + 7. Step 2: Find the differential du. Differentiate u with respect to x: (du)/(dx) = (d)/(dx)(3x^3 + 7) = 9x^2 So, du = 9x^2 \, dx. Step 3: Express x^2 \, dx in terms of du. From du = 9x^2 \, dx, we have x^2 \, dx = (1)/(9) du. Step 4: Rewrite the integral in terms of u. The original integral is (x^2 \, dx)/([5]3x^3 + 7). Substitute u and du: (1)/(9) du[5]u = (1)/(9) (1)/(u^1/5) du Rewrite the term with a negative exponent: (1)/(9) u^-1/5 du Step 5: Integrate with respect to u. Use the power rule for integration, u^n \, du = u^n+1n+1 + C: (1)/(9) ( u^-1/5 + 1-1/5 + 1 ) + C (1)/(9) ( u^4/54/5 ) + C (1)/(9) ( (5)/(4) u^4/5 ) + C (5)/(36) u^4/5 + C Step 6: Substitute back u = 3x^3 + 7. (5)/(36) (3x^3 + 7)^4/5 + C The final answer is (5)/(36) (3x^3 + 7)^4/5 + C.