What is the integral of (2x^4+5x^2+1)/(x^3+x) ?

The integral of (2x^4+5x^2+1)/(x^3+x) is x^2+ln|x^2+1|+ln|x|+1+C

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integral of (2x^4+5x^2+1)/(x^3+x)

\int \frac{2 x ^{4} + 5 x ^{2} + 1}{x ^{3} + x} d x

x^{2} + ln x^{2} + 1 + ln ∣ x ∣ + 1 + C

Solution steps

\int \frac{2 x ^{4} + 5 x ^{2} + 1}{x ^{3} + x} d x

Expand

\frac{2 x ^{4} + 5 x ^{2} + 1}{x ^{3} + x} : \frac{2 x ^{4}}{x ^{3} + x} + \frac{5 x ^{2}}{x ^{3} + x} + \frac{1}{x ^{3} + 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 \frac{2 x ^{4}}{x ^{3} + x} d x + \int \frac{5 x ^{2}}{x ^{3} + x} d x + \int \frac{1}{x ^{3} + x} d x

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

\int \frac{5 x ^{2}}{x ^{3} + x} d x = \frac{5}{2} ln x^{2} + 1

\int \frac{1}{x ^{3} + x} d x = ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

= x^{2} + 1 - ln x^{2} + 1 + \frac{5}{2} ln x^{2} + 1 + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

Simplify

x^{2} + 1 - ln x^{2} + 1 + \frac{5}{2} ln x^{2} + 1 + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 : x^{2} + ln x^{2} + 1 + ln ∣ x ∣ + 1

= x^{2} + ln x^{2} + 1 + ln ∣ x ∣ + 1

Add a constant to the solution

= x^{2} + ln x^{2} + 1 + ln ∣ x ∣ + 1 + C

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What is the integral of (2x^4+5x^2+1)/(x^3+x) ?
The integral of (2x^4+5x^2+1)/(x^3+x) is x^2+ln|x^2+1|+ln|x|+1+C