What is the integral of ((2x^5-4sqrt(x)+8x)/x) ?

The integral of ((2x^5-4sqrt(x)+8x)/x) is (2x^5)/5-8sqrt(x)+8x+C

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integral of ((2x^5-4sqrt(x)+8x)/x)

\int (\frac{2 x ^{5} - 4 x + 8 x}{x}) d x

\frac{2 x ^{5}}{5} - 8 x + 8 x + C

Solution steps

\int \frac{2 x ^{5} - 4 x + 8 x}{x} d x

Expand

\frac{2 x ^{5} - 4 x + 8 x}{x} : 2 x^{4} - \frac{4}{x} + 8

= \int 2 x^{4} - \frac{4}{x} + 8 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

= \int 2 x^{4} d x - \int \frac{4}{x} d x + \int 8 d x

\int 2 x^{4} d x = \frac{2 x ^{5}}{5}

\int \frac{4}{x} d x = 8 x

\int 8 d x = 8 x

= \frac{2 x ^{5}}{5} - 8 x + 8 x

Add a constant to the solution

= \frac{2 x ^{5}}{5} - 8 x + 8 x + C

x \to + 0 lim (x \cdot + \frac{1}{x})

\frac{d}{d x} (e^{c o s h (2 x)})

x \to 1 + lim (f (x))

\frac{ln ( cos ( 2 x ) )}{5 x}

\frac{( x - 9 )}{( 5 x - 1 )}, at x = 1

What is the integral of ((2x^5-4sqrt(x)+8x)/x) ?
The integral of ((2x^5-4sqrt(x)+8x)/x) is (2x^5)/5-8sqrt(x)+8x+C