What is the integral of (-4x^5-x^{3/4}-2)/(9x^{4/3)} ?

The integral of (-4x^5-x^{3/4}-2)/(9x^{4/3)} is (2(-5x^5-14x^{3/4}+35))/(105x^{1/3)}+C

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integral of (-4x^5-x^{3/4}-2)/(9x^{4/3)}

\int \frac{- 4 x ^{5} - x ^{\frac{3}{4}} - 2}{9 x ^{\frac{4}{3}}} d x

\frac{2 ( - 5 x ^{5} - 14 x ^{\frac{3}{4}} + 35 )}{105 x ^{\frac{1}{3}}} + C

Solution steps

\int \frac{- 4 x ^{5} - x ^{\frac{3}{4}} - 2}{9 x ^{\frac{4}{3}}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{9} \cdot \int \frac{- 4 x ^{5} - x ^{\frac{3}{4}} - 2}{x ^{\frac{4}{3}}} d x

Expand

\frac{- 4 x ^{5} - x ^{\frac{3}{4}} - 2}{x ^{\frac{4}{3}}} : - 4 x^{\frac{11}{3}} - \frac{1}{x ^{\frac{7}{12}}} - \frac{2}{x ^{\frac{4}{3}}}

= \frac{1}{9} \cdot \int - 4 x^{\frac{11}{3}} - \frac{1}{x ^{\frac{7}{12}}} - \frac{2}{x ^{\frac{4}{3}}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{9} (- \int 4 x^{\frac{11}{3}} d x - \int \frac{1}{x ^{\frac{7}{12}}} d x - \int \frac{2}{x ^{\frac{4}{3}}} d x)

\int 4 x^{\frac{11}{3}} d x = \frac{6}{7} x^{\frac{14}{3}}

\int \frac{1}{x ^{\frac{7}{12}}} d x = \frac{12}{5} x^{\frac{5}{12}}

\int \frac{2}{x ^{\frac{4}{3}}} d x = - \frac{6}{x ^{\frac{1}{3}}}

= \frac{1}{9} (- \frac{6}{7} x^{\frac{14}{3}} - \frac{12}{5} x^{\frac{5}{12}} - (- \frac{6}{x ^{\frac{1}{3}}}))

Simplify

\frac{1}{9} (- \frac{6}{7} x^{\frac{14}{3}} - \frac{12}{5} x^{\frac{5}{12}} - (- \frac{6}{x ^{\frac{1}{3}}})) : \frac{2 ( - 5 x ^{5} - 14 x ^{\frac{3}{4}} + 35 )}{105 x ^{\frac{1}{3}}}

= \frac{2 ( - 5 x ^{5} - 14 x ^{\frac{3}{4}} + 35 )}{105 x ^{\frac{1}{3}}}

Add a constant to the solution

= \frac{2 ( - 5 x ^{5} - 14 x ^{\frac{3}{4}} + 35 )}{105 x ^{\frac{1}{3}}} + C

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What is the integral of (-4x^5-x^{3/4}-2)/(9x^{4/3)} ?
The integral of (-4x^5-x^{3/4}-2)/(9x^{4/3)} is (2(-5x^5-14x^{3/4}+35))/(105x^{1/3)}+C