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

The integral of (8x^5+4x)/(2x^2) is x^4+2ln|x|+C

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

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

x^{4} + 2 ln ∣ x ∣ + C

Solution steps

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

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{8 x ^{5} + 4 x}{x ^{2}} d x

Expand

\frac{8 x ^{5} + 4 x}{x ^{2}} : 8 x^{3} + \frac{4}{x}

= \frac{1}{2} \cdot \int 8 x^{3} + \frac{4}{x} 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}{2} (\int 8 x^{3} d x + \int \frac{4}{x} d x)

\int 8 x^{3} d x = 2 x^{4}

\int \frac{4}{x} d x = 4 ln ∣ x ∣

= \frac{1}{2} (2 x^{4} + 4 ln ∣ x ∣)

Simplify

\frac{1}{2} (2 x^{4} + 4 ln ∣ x ∣) : x^{4} + 2 ln ∣ x ∣

= x^{4} + 2 ln ∣ x ∣

Add a constant to the solution

= x^{4} + 2 ln ∣ x ∣ + C

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