Expert answers to "Evaluate the integrals ∫181x−13(1−x23)32dx"

necessaryh

Answered question

2021-10-22

Evaluate the integrals

\int_{\frac{1}{8}}^{1} x^{- \frac{1}{3}} {(1 - x^{\frac{2}{3}})}^{\frac{3}{2}} d x

Delorenzoz

Skilled2021-10-23Added 91 answers

Given:

\int_{\frac{1}{8}}^{1} x^{- \frac{1}{3}} {(1 - x^{\frac{2}{3}})}^{\frac{3}{2}} d x

Calculation:

\int_{\frac{1}{8}}^{1} x^{- \frac{1}{3}} {(1 - x^{\frac{2}{3}})}^{\frac{3}{2}} d x

put

1 - x^{\frac{2}{3}} = u \Rightarrow - \frac{2}{3} x^{- \frac{1}{3}} d x = d u

When

x = \frac{1}{8} \Rightarrow u = 1 - {(\frac{1}{8})}^{\frac{2}{3}} = 1 - \frac{1}{4} = \frac{3}{4}

when

x = 1 \Rightarrow u = 1 - 1 = 0

\frac{- 3}{2} \int_{\frac{3}{4}}^{0} u^{\frac{3}{2}} d u

= \frac{- 3}{2} {[0 - (\frac{3}{4})]}_{\frac{3}{4}}^{0}

= \frac{- 3}{5} \times \frac{- 9 \sqrt{3}}{32}

= \frac{27 \sqrt{3}}{160}

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