Question

Find the following antiderivative: ∫4x⁴(–x^5 + 8)^6 dx. Provide your answer below:

Answer 1

`\displaystyle -(4)/(35)(8-x^5)^7}+C`

Explanation:

`\displaystyle \int4x^4(-x^5+8)^6dx`

Let
`u=-x^5+8` and
`du=-5x^4dx` so that
`\displaystyle -(4)/(5)du=4x^4dx`:

`\displaystyle -(4)/(5)\int u^6du\\\\=-(4)/(5)\biggr((u^7)/(7)\biggr)+C\\\\=-(4)/(5)\biggr(((-x^5+8)^7)/(7)\biggr)+C\\\\=-(4)/(35)(8-x^5)^7}+C`