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
let's solve this integral. Evaluate x^5 (x^3+1)^1/2 dx. Step 1: Use substitution. Let u = x^3+1. Then, differentiate u with respect to x: du = 3x^2 dx. From this, we can express x^2 dx as (1)/(3) du. Also, from u = x^3+1, we can write x^3 = u-1. Step 2: Rewrite the integral in terms of u. The integral can be written as x^3 · x^2 (x^3+1)^1/2 dx. Substitute the expressions from Step 1: (u-1) u^1/2 ((1)/(3) du) = (1)/(3) (u-1)u^1/2 du Distribute u^1/2 inside the parenthesis: = (1)/(3) (u^1 · u^1/2 - 1 · u^1/2) du = (1)/(3) (u^3/2 - u^1/2) du Step 3: Integrate term by term using the power rule y^n dy = y^n+1n+1 + C. = (1)/(3) (u^3/2+13/2+1 - u^1/2+11/2+1) + C = (1)/(3) (u^5/25/2 - u^3/23/2) + C Simplify the fractions: = (1)/(3) ((2)/(5)u^5/2 - (2)/(3)u^3/2) + C Distribute the (1)/(3): = (2)/(15)u^5/2 - (2)/(9)u^3/2 + C Step 4: Substitute back u = x^3+1 to express the result in terms of x. = (2)/(15)(x^3+1)^5/2 - (2)/(9)(x^3+1)^3/2 + C The final answer is (2)/(15)(x^3+1)^5/2 - (2)/(9)(x^3+1)^3/2 + C. What's next? Send 'em! 📸