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Answer 1 · 2017-05-12T00:27:27

$= \frac{8 \sqrt{x}}{3} - \frac{\sqrt{3}}{3 x^{3}} + \frac{5}{12 x^{2}} + \frac{4 x^{5}}{35} + C$ $=(8sqrtx)/3 - sqrt3/(3x^3) + 5/(12x^2) + (4x^5)/35 +C$

Explanation:

$\int (\frac{4}{3} x^{- \frac{1}{2}} + \sqrt{3} x^{- 4} - \frac{5}{6 x^{3}} + \frac{4}{7 x^{- 4}}) d x$ $int(4/3x^(-1/2)+sqrt3x^(-4)-5/(6x^3)+4/(7x^(-4)))dx$

Rewrite with no $x$ $x$ 's in the denominators:
$= \int (\frac{4}{3} x^{- \frac{1}{2}} + \sqrt{3} x^{- 4} - \frac{5}{6} x^{- 3} + \frac{4}{7} x^{4}) d x$ $=int( 4/3 x^(-1/2)+sqrt3 x^(-4)-5/6 x^(-3)+4/7 x^4)dx$

Integrate using the "opposite of" the differentiation power rule:

$= \frac{8}{3} x^{\frac{1}{2}} - \frac{\sqrt{3}}{3} x^{- 3} + \frac{5}{12} x^{- 2} + \frac{4}{35} x^{5} + C$ $=8/3x^(1/2)-sqrt3/3 x^(-3)+5/12x^(-2)+4/35x^5 +C$

For the sake of better notation, only use positive exponents:
$= \frac{8 \sqrt{x}}{3} - \frac{\sqrt{3}}{3 x^{3}} + \frac{5}{12 x^{2}} + \frac{4 x^{5}}{35} + C$ $=(8sqrtx)/3 - sqrt3/(3x^3) + 5/(12x^2) + (4x^5)/35 +C$

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