What is the integral of (x^8+x^4-x^2-7)/x ?

The integral of (x^8+x^4-x^2-7)/x is (x^8)/8+(x^4)/4-(x^2)/2-7ln|x|+C

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integral of (x^8+x^4-x^2-7)/x

\int \frac{x ^{8} + x ^{4} - x ^{2} - 7}{x} d x

\frac{x ^{8}}{8} + \frac{x ^{4}}{4} - \frac{x ^{2}}{2} - 7 ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{8} + x ^{4} - x ^{2} - 7}{x} d x

Expand

\frac{x ^{8} + x ^{4} - x ^{2} - 7}{x} : x^{7} + x^{3} - x - \frac{7}{x}

= \int x^{7} + x^{3} - x - \frac{7}{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

= \int x^{7} d x + \int x^{3} d x - \int x d x - \int \frac{7}{x} d x

\int x^{7} d x = \frac{x ^{8}}{8}

\int x^{3} d x = \frac{x ^{4}}{4}

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

\int \frac{7}{x} d x = 7 ln ∣ x ∣

= \frac{x ^{8}}{8} + \frac{x ^{4}}{4} - \frac{x ^{2}}{2} - 7 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{8}}{8} + \frac{x ^{4}}{4} - \frac{x ^{2}}{2} - 7 ln ∣ x ∣ + C

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What is the integral of (x^8+x^4-x^2-7)/x ?
The integral of (x^8+x^4-x^2-7)/x is (x^8)/8+(x^4)/4-(x^2)/2-7ln|x|+C