Question

Find an anti derivative for each function when C = 0

Answer 1

Given:

`(9)/(8)\sqrt[8]{x}`

You can find the antiderivative by integrating it:

1. Set up:

`\int(9)/(8)\sqrt[8]{x}\text{ }dx`

2. You can rewrite it in this form:

`=(9)/(8)\int x^{(1)/(8)}dx`

3. Apply this Integration Rule:

`\int x^ndx=(x^(n+1))/(n+1)`

Then, you get:

`=(9)/(8)(\frac{x^{(1)/(8)+1}}{(1)/(8)+1})+C`
`=(9)/(8)(\frac{x^{(9)/(8)}}{(9)/(8)})+C`

4. Simplify:

`=(9)/(8)(\frac{x^{(1)/(8)+1}}{(1)/(8)+1})+C`
`=(9)/(8)(\frac{8\sqrt[8]{x^9}}{9})+C`
`=\sqrt[8]{x^9}+C`

Remember this Property for Radicals:

`\sqrt[m]{b^n}=b^{(n)/(m)}`

You can rewrite the expression in this form:

`=\sqrt[8]{x\cdot x^8}+C`

Applying this Property for Radicals:

`\sqrt[n]{b^n}=b`

You get:

`=x\sqrt[8]{x}+C`

5. Knowing that:

`C=0`

You obtain:

`=x\sqrt[8]{x}`

Hence, the answer is:

`=x\sqrt[8]{x}`