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
To integrate x^3 sqrt(x^2+1) \, dx, we can use a substitution method. Step 1: Set up the substitution. Let u = x^2+1. Then, differentiate u with respect to x: (du)/(dx) = 2x This means du = 2x \, dx, or x \, dx = (1)/(2) du. Also, from u = x^2+1, we can express x^2 as x^2 = u-1. Step 2: Rewrite the integral in terms of u. The original integral is x^3 sqrt(x^2+1) \, dx. We can rewrite x^3 as x^2 · x. So, the integral becomes x^2 sqrt(x^2+1) · x \, dx. Substitute x^2 = u-1, sqrt(x^2+1) = sqrt(u), and x \, dx = (1)/(2) du: (u-1) sqrt(u) · (1)/(2) du (1)/(2) (u-1) u^1/2 du Distribute u^1/2: (1)/(2) (u^3/2 - u^1/2) du Step 3: Integrate with respect to u. Apply the power rule for integration, u^n \, du = u^n+1n+1 + C: (1)/(2) ( u^3/2+13/2+1 - u^1/2+11/2+1 ) + C (1)/(2) ( u^5/25/2 - u^3/23/2 ) + C (1)/(2) ( (2)/(5) u^5/2 - (2)/(3) u^3/2 ) + C (1)/(5) u^5/2 - (1)/(3) u^3/2 + C Step 4: Substitute back u = x^2+1. (1)/(5) (x^2+1)^5/2 - (1)/(3) (x^2+1)^3/2 + C Step 5: Factor and simplify the expression. Factor out the common term (x^2+1)^3/2: (x^2+1)^3/2 ( (1)/(5) (x^2+1) - (1)/(3) ) + C Find a common denominator for the terms inside the parenthesis: (x^2+1)^3/2 ( (3(x^2+1) - 5)/(15) ) + C (x^2+1)^3/2 ( (3x^2+3 - 5)/(15) ) + C (x^2+1)^3/2 ( (3x^2-2)/(15) ) + C (1)/(15) (3x^2-2) (x^2+1)^3/2 + C Send me the next one 📸