1) The integral of sin^3(5x) dx is not easily solvable using elementary functions. You may need to use trigonometric identities or special techniques like integration by parts.
2) The integral of 5√x - 2x^3 + 4/x^2 dx can be solved by integrating each term separately: ∫5√x dx = 5∫x^(1/2) dx = 5(x^(3/2))/(3/2) + C = (10/3)x^(3/2) + C ∫-2x^3 dx = -2∫x^3 dx = -2(x^4/4) + C = -1/2(x^4) + C ∫4/x^2 dx = 4∫x^(-2) dx = 4(-1/x) + C = -4/x + C
Therefore, the solution to the integral 5√x - 2x^3 + 4/x^2 dx is (10/3)x^(3/2) - 1/2(x^4) - 4/x + C.
3) The integral of (2x + 3)/√(2x^2 - x + 6) dx can be solved by first simplifying the expression under the square root. The integrand can be rewritten as (2x + 3)/√(2(x^2 - x/2) + 6). Upon rearranging, the integrand transforms into [2(x - 1/4) + 3 + 1/2]/√(2(x - 1/4)^2 + 23/8).
After substitution into that form, the integral will be solvable using techniques such as trigonometric substitutions or partial fraction decomposition.
1) The integral of sin^3(5x) dx is not easily solvable using elementary functions. You may need to use trigonometric identities or special techniques like integration by parts.
2) The integral of 5√x - 2x^3 + 4/x^2 dx can be solved by integrating each term separately:
∫5√x dx = 5∫x^(1/2) dx = 5(x^(3/2))/(3/2) + C = (10/3)x^(3/2) + C
∫-2x^3 dx = -2∫x^3 dx = -2(x^4/4) + C = -1/2(x^4) + C
∫4/x^2 dx = 4∫x^(-2) dx = 4(-1/x) + C = -4/x + C
Therefore, the solution to the integral 5√x - 2x^3 + 4/x^2 dx is (10/3)x^(3/2) - 1/2(x^4) - 4/x + C.
3) The integral of (2x + 3)/√(2x^2 - x + 6) dx can be solved by first simplifying the expression under the square root. The integrand can be rewritten as (2x + 3)/√(2(x^2 - x/2) + 6).
Upon rearranging, the integrand transforms into [2(x - 1/4) + 3 + 1/2]/√(2(x - 1/4)^2 + 23/8).
After substitution into that form, the integral will be solvable using techniques such as trigonometric substitutions or partial fraction decomposition.