We need to rewrite the integral so that it is in the form of a product.
∫(u2 + u)(2u3 + 3u2 + 1)-2/3 du
We can use the substitution method.
Let x = 2u3 + 3u2 + 1
dx = (6u2 + 6u)du
dx = 6(u2 + u)du
(1/6)dx = (u2 + u)du
Using these substituted terms, we can write an equivalent integral that is easier to evaluate.
(1/6)∫x-2/3 dx =
(1/6)(3)x1/3 + C =
(1/2)x1/3 + C
Now we substitute the value of x back into the integral so that it is in terms of u.
(1/2)(2u3 + 3u2 + 1)1/3 + C
To check, if we take the derivative of the solution, we should get back the original expression that we need to integrate.