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.