Given:
[tex]\frac{9}{8}\sqrt[8]{x}[/tex]
You can find the antiderivative by integrating it:
1. Set up:
[tex]\int\frac{9}{8}\sqrt[8]{x}\text{ }dx[/tex]
2. You can rewrite it in this form:
[tex]=\frac{9}{8}\int x^{\frac{1}{8}}dx[/tex]
3. Apply this Integration Rule:
[tex]\int x^ndx=\frac{x^{n+1}}{n+1}[/tex]
Then, you get:
[tex]=\frac{9}{8}(\frac{x^{\frac{1}{8}+1}}{\frac{1}{8}+1})+C[/tex][tex]=\frac{9}{8}(\frac{x^{\frac{9}{8}}}{\frac{9}{8}})+C[/tex]
4. Simplify:
[tex]=\frac{9}{8}(\frac{x^{\frac{1}{8}+1}}{\frac{1}{8}+1})+C[/tex][tex]=\frac{9}{8}(\frac{8\sqrt[8]{x^9}}{9})+C[/tex][tex]=\sqrt[8]{x^9}+C[/tex]
Remember this Property for Radicals:
[tex]\sqrt[m]{b^n}=b^{\frac{n}{m}}[/tex]
You can rewrite the expression in this form:
[tex]=\sqrt[8]{x\cdot x^8}+C[/tex]
Applying this Property for Radicals:
[tex]\sqrt[n]{b^n}=b[/tex]
You get:
[tex]=x\sqrt[8]{x}+C[/tex]
5. Knowing that:
[tex]C=0[/tex]
You obtain:
[tex]=x\sqrt[8]{x}[/tex]
Hence, the answer is:
[tex]=x\sqrt[8]{x}[/tex]
