\(\int_{0}^{8}\displaystyle (x^{\frac{2}{3}}+1)\,dx\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{136}{5}\)
By the power rule, \(\int x^{2/3}\,dx = \frac{3}{5}x^{5/3}\):
\( \displaystyle \int_{0}^{8}\left(x^{\frac{2}{3}}+1\right)dx = \left[\frac{3}{5}x^{\frac{5}{3}} + x\right]_{0}^{8} \)
\( \displaystyle = \frac{3}{5}(8)^{\frac{5}{3}} + 8 \)
Since \(8^{1/3} = 2\), \(8^{5/3} = 2^5 = 32\):
\( \displaystyle = \frac{96}{5} + 8 = \frac{136}{5} \)