integrate (4x^3 +3)(x^4 +3x +16)^2 dx

For this integration, we will use integration by substitution:First, we will let substitute u = X⁴ + 3X + 16 (this may be known as g(x) function and we will need to find the derivative of our u/g(x) for this , we can easily derive that du/dx = 4X³ +3 (g'(x) = 4X³ +3)and we can then rearrange the du/dx equation to have du = (4X³ +3) dx. This can be found in our original integral, (4X³ +3) dx hence rearrange the integral to integrate with respect to u (du) which gets to integral of U² du and we can apply the easy rule of U^power+1/power+1 to find that is U³/3 +CSubstitute the original u , we will get answer ( X⁴ + 3X + 16)³/3 +c