"Evaluate the following definite integral. ∫1681(2x4−3x45)dx" was answered by our community

Ramsey

Answered question

2021-09-26

Evaluate the following definite integral.
$\int_{16}^{81} (2 \sqrt[4]{x} - \frac{3}{{\sqrt[4]{x}}^{5}}) d x$

Answer & Explanation

Mayme

Skilled2021-09-27Added 103 answers

$\int_{16}^{81} (2 \sqrt[4]{x} - \frac{3}{{\sqrt[4]{x}}^{5}}) d x = \int_{16}^{81} (2 (x^{\frac{1}{4}}) - 3 (x^{- \frac{5}{4}})) d x$
$= 2 (\frac{x^{\frac{1}{4} + 1}}{(\frac{1}{4} + 1)}) - 3 (\frac{x^{- \frac{5}{4}}}{- \frac{5}{4} + 1}) ∣_{16}^{81}$
$= 2 (\frac{x^{\frac{5}{4}}}{(\frac{5}{4})}) - 3 (\frac{x^{- \frac{1}{4}}}{- \frac{1}{4}}) ∣_{16}^{81}$
$= 2 (\frac{4}{5}) (x^{\frac{5}{4}}) - 3 (- \frac{4}{1}) (x^{- \frac{1}{4}}) ∣_{16}^{81}$
$= \frac{8}{5} x^{\frac{5}{4}} + 12 x^{- \frac{1}{4}} ∣_{16}^{81}$
$= (\frac{8}{5} {(81)}^{\frac{5}{4}} + 12 {(81)}^{- \frac{1}{4}}) - (\frac{8}{5} {(16)}^{\frac{5}{4}} + 12 {(16)}^{- \frac{1}{4}})$
$= 388.8 + 4 - 51.2 - 6 = 335.6$

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