Question

How do you find the antiderivative of `int (2x^3-1)(x^4-2x)^6dx` from `[-1,1]`?

Answer 1

`int_(-1)^1(2x^3-1)(x^4-2x)^6dx=(-1094)/7`

Explanation:

`int_(-1)^1(2x^3-1)(x^4-2x)^6dx`

We will apply the substitution `u=x^4-2x`. Differentiating this reveals that `du=(4x^3-2)dx`. Note that the `(2x^3-1)` term is exactly half this, so we can modify the integral as follows:

`=1/2int_(-1)^1(x^4-2x)^6(4x^3-2)dx`

Now we can substitute in our `u` and `du` values. However, we also should remember to change our bounds! Take `-1` and `1` and plug them into our `u=x^4-2x` term. That is, the bound of `-1` becomes `(-1)^4-2(-1)=3` and the bound of `1` becomes `1^4-2(1)=-1`. Thus:

`=1/2int_3^(-1)u^6du=1/2[u^7/7]_3^(-1)=1/2[(-1)^7/7-3^7/7]`

`=1/2[(-2188)/7]=(-1094)/7`