∫4x3⋅(x4+2)6dx

Sure, let's solve the integral step by step:

1. First, we notice that the integral is in the form of a function and its derivative. We can rewrite the integral as ∫u'v dx, where u = x^4 + 2 and v = (x^4 + 2)^6.

2. We can use the substitution method to solve this integral. Let's set u = x^4 + 2. Then, du = 4x^3 dx.

3. Substitute u and du into the integral: ∫du * u^6.

4. Now, we can use the power rule for integration, which states ∫x^n dx = (1/(n+1))x^(n+1) + C, where C is the constant of integration.

5. Applying the power rule, we get (1/7)u^7 + C.

6. Substitute u back into the equation to get the final answer: (1/7)(x^4 + 2)^7 + C.