What is the antiderivative of the cube root of (x^2)+2
To find the antiderivative of the cube root of (x^2)+2, we can follow these steps:
1. Let's rewrite the given expression as (x^2+2)^(1/3).
2. The antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x). In other words, F(x) is the function whose derivative is equal to f(x).
3. To find the antiderivative of (x^2+2)^(1/3), we can use a substitution because the function inside the cube root is a composite function.
4. Let's substitute u = x^2+2. This gives us du = 2x dx (using the chain rule).
5. Solving for dx, we have dx = du/(2x).
6. Substituting dx and (x^2+2)^(1/3) with the corresponding expressions in terms of u, we have:
∫(x^2+2)^(1/3) dx = ∫u^(1/3) * (du/(2x))
7. Simplifying the expression inside the integral, we have (1/2)∫(u^(1/3) / x) du.
8. Now we can separate the variables and integrate:
(1/2)∫(u^(1/3)/x) du = (1/2)∫u^(1/3) * (1/x) du.
9. We can pull out the constant (1/2) and (1/x) since they do not depend on u:
(1/2) * (1/x) * ∫u^(1/3) du.
10. Integrating u^(1/3) with respect to u gives us:
(1/2) * (1/x) * (3/4) * u^(4/3) + C, where C is the constant of integration.
11. Finally, substituting back u = x^2+2, we have:
(1/2) * (1/x) * (3/4) * (x^2+2)^(4/3) + C.
Therefore, the antiderivative of the cube root of (x^2)+2 is (1/2) * (1/x) * (3/4) * (x^2+2)^(4/3) + C, where C is the constant of integration.