Question

Evaluate ${\int }_0^a\left[x^n\right]dx$.

• A. $\left(\sum _{r=1}^K(r-1)[r^{\frac{1}{n}}-{(r-1)}^{\frac{1}{n}}]\right)+K(a-K^{\frac{1}{n}})$
• B. $\left(\sum _{r=1}^K(r+1)[r^{\frac{1}{n}}-{(r+1)}^{\frac{1}{n}}]\right)+K(a-K^{\frac{1}{n}})$
• C. $\left(\sum _{r=1}^K(r-1{)}^2[r^{\frac{1}{n}}+{(r-1)}^{\frac{1}{n}}]\right)-K(a-K^{\frac{1}{n}})$
• D. None of these

Answer 1

The correct option is A $\left(\sum _{r=1}^K(r-1)[r^{\frac{1}{n}}-{(r-1)}^{\frac{1}{n}}]\right)+K(a-K^{\frac{1}{n}})$

Let $K$ be a natural number such that $K\le a^n\le K+1$
Then, ${\int }_0^a\left[x^n\right]dx={\int }_0^1\left[x^n\right]dx+{\int }_1^{2^{\frac{1}{n}}}\left[x^n\right]dx+{\int }_{2^{\frac{1}{n}}}^{3^{\frac{1}{n}}}\left[x^n\right]dx+\cdots +{\int }_{{(K-1)}^{\frac{1}{n}}}^{K^{\frac{1}{n}}}\left[x^n\right]dx+{\int }_{K^{\frac{1}{n}}}^{{\left(a^n\right)}^{\frac{1}{n}}}\left[x^n\right]dx$
$={\int }_0^{1}0dx+{\int }_1^{2^{\frac{1}{n}}}1dx+{\int }_{2^{\frac{1}{n}}}^{3^{\frac{1}{n}}}2dx+\cdots +{\int }_{{(K-1)}^{\frac{1}{n}}}^{K^{\frac{1}{n}}}(K-1)dx+{\int }_{K^{\frac{1}{n}}}^{{\left(a^n\right)}^{\frac{1}{n}}}Kdx$
$=0+1.(2^{\frac{1}{n}}-1)+2.(3^{\frac{1}{n}}-2^{\frac{1}{n}})+\cdots +(K-1)(K^{\frac{1}{n}}-{(K-1)}^{\frac{1}{n}})+K(a-K^{\frac{1}{n}})$
$⟹\left(\sum _{r=1}^K(r-1)[r^{\frac{1}{n}}-{(r-1)}^{\frac{1}{n}}]\right)+K(a-K^{\frac{1}{n}})$
Ans: A