If-I-1-8-1-x-2-3-x-dx-then-what-is-the-value-of-81e-I-

Question Number 167037 by HongKing last updated on 05/Mar/22

If I = \int_{- 1}^{- 8} \frac{1}{x^{\frac{2}{3}} - x} dx

then what is the value of 81 e^{I} ?

Answered by ghakhan last updated on 05/Mar/22

I = {[- 3 \ln ∣ \sqrt[3]{x} - 1 ∣]}_{- 1}^{- 8}

= - 3 (\ln (3) - \ln (2)) = - 3 \ln (\frac{3}{2})

{81 e}^{I} = {81 e}^{\ln {(\frac{3}{2})}^{- 3}} = \frac{81 \cdot 8}{27} = 24

Answered by qaz last updated on 05/Mar/22

I = \int_{- 1}^{- 8} \frac{dx}{x^{2 / 3} - x} = - \int_{1}^{8} \frac{dx}{x^{2 / 3} + x} = - \int_{1}^{8} \frac{dx}{x^{2 / 3} (1 + x^{1 / 3})}

= - 3 \int_{1}^{8} \frac{d (x^{1 / 3})}{1 + x^{1 / 3}} = 3 \ln (1 + x^{1 / 3}) ∣_{8}^{1} = \ln \frac{8}{27}

\Rightarrow {81 e}^{I} = {81 e}^{\ln \frac{8}{27}} = 81 \times \frac{8}{27} = 24

Facebook Tweet Pin