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Find an anti derivative for each function when C = 0

Find an anti derivative for each function when C = 0-example-1
User PaulMiami
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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}
User James Gan
by
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