Simplify and integrate the following: integral from [((x2 - 5/x) * x^(-1/3) + 2sec2 x + 5 sin 3x - 6/x)] dx

The problem asks to simplify and evaluate the indefinite integral: ( ( x^2 - (5)/(x) ) x^-1/3 + 2 ^2 x + 5 3x - (6)/(x) ) dx Step 1: Simplify the integrand by distributing x^-1/3 and combining terms. ( x^2 - (5)/(x) ) x^-1/3 = x^2 · x^-1/3 - (5)/(x) · x^-1/3 Using the exponent rule a^m · a^n = a^m+n: x^2 - 1/3 - 5 x^-1 - 1/3 = x^6/3 - 1/3 - 5 x^-3/3 - 1/3 = x^5/3 - 5 x^-4/3 So the integral becomes: ( x^5/3 - 5 x^-4/3 + 2 ^2 x + 5 3x - (6)/(x) ) dx Step 2: Integrate each term separately using the standard integration rules: • x^n dx = x^n+1n+1 + C (for n ≠ -1) • (1)/(x) dx = |x| + C • ^2 x dx = x + C • (ax) dx = -(1)/(a) (ax) + C Step 3: Apply the integration rules to each term. For x^5/3: x^5/3 dx = x^5/3+15/3+1 = x^8/38/3 = (3)/(8) x^8/3 For -5 x^-4/3: -5 x^-4/3 dx = -5 x^-4/3+1-4/3+1 = -5 x^-1/3-1/3 = 15 x^-1/3 For 2 ^2 x: 2 ^2 x dx = 2 x For 5 3x: 5 3x dx = 5 ( -(1)/(3) 3x ) = -(5)/(3) 3x For -(6)/(x): -(6)/(x) dx = -6 |x| Step 4: Combine all the integrated terms and add the constant of integration, C. (3)/(8) x^8/3 + 15 x^-1/3 + 2 x - (5)/(3) 3x - 6 |x| + C The final answer is: (3)/(8) x^8/3 + 15 x^-1/3 + 2 x - (5)/(3) 3x - 6 |x| + C That's 2 down. 3 left today — send the next one.