Evaluate the following integral:

(X)(cube root(x+1))dx

9x+1)^4/4+c

(x+1)^4/4+c *

To evaluate the integral ∫x∛(x+1)dx, we can use integration by substitution. Let's use u = x + 1 as the substitution variable.

First, let's find the derivative of u with respect to x:

du/dx = 1

Now, solve for dx in terms of du:

dx = du

Next, substitute x + 1 for u and dx for du in the integral:

∫x∛(x+1)dx = ∫(u-1)∛udu

To simplify the expression, let's split the integral:

∫(u-1)∛udu = ∫u∛udu - ∫∛udu

Now, we can integrate each term separately:

∫u∛udu = ∫u^(4/3)du
= (3/7)u^(7/3) + C

∫∛udu = (3/4)u^(4/3) + C

Now, substitute back u = x + 1:

= (3/7)(x+1)^(7/3) - (3/4)(x+1)^(4/3) + C

Therefore, the value of the integral ∫x∛(x+1)dx is (3/7)(x+1)^(7/3) - (3/4)(x+1)^(4/3) + C, where C is the constant of integration.