integrate (27x -9x^2)/sqrt(x^3 +9)
To integrate the function (27x - 9x^2) / sqrt(x^3 + 9), we can use a method called substitution.
Let's start by choosing a substitution variable. In this case, we can let u be equal to x^3 + 9.
Now, let's find du/dx by differentiating both sides of the equation u = x^3 + 9 with respect to x. We get:
du/dx = 3x^2
To find dx, we can rearrange this equation as follows:
dx = du / (3x^2)
Now, let's substitute these expressions into the integral:
∫ (27x - 9x^2) / √(x^3 + 9) dx
∫ (27x - 9x^2) / √u * (du / 3x^2)
Next, let's simplify the expression:
∫ 3(9 - 3x) / (3x^2) * √u du
∫ (9 - 3x) / (x^2) * √u du
Now, we can integrate each term separately.
The integral of √u du can be expressed as (2/3) * u^(3/2) + C1, where C1 is the constant of integration.
The integral of (9 - 3x) / (x^2) can be approached using long division. Dividing 9 - 3x by x^2 gives:
9 - 3x = 0 * x^2 + (-3/x) * x^2 + 9/x^2
So, the integral of (9 - 3x) / (x^2) is equal to ∫ (-3/x) dx + 9 * ∫ (1/x^2) dx.
The first term, ∫ (-3/x) dx, can be evaluated as -3 * ln|x| + C2, where C2 is another constant of integration.
The second term, ∫ (1/x^2) dx, simplifies to -1/x + C3, where C3 is the constant of integration.
Now, substituting all the results back into the original expression, we get:
∫ (27x - 9x^2) / √(x^3 + 9) dx
= 3(2/3) * (x^3 + 9)^(3/2) + C1 + (-3 * ln|x| + C2) + 9 * (-1/x + C3)
Simplifying further, we have:
= 2(x^3 + 9)^(3/2) + C1 - 3ln|x| + C2 - 9/x + C3
Therefore, the final result of the integration is:
2(x^3 + 9)^(3/2) - 3ln|x| - 9/x + (C1 + C2 + C3)
So, integrating the function (27x - 9x^2) / sqrt(x^3 + 9) gives us 2(x^3 + 9)^(3/2) - 3ln|x| - 9/x + (C1 + C2 + C3), where C1, C2, and C3 are constants of integration.