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

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

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

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

\frac{1}{2} ln x^{4} + 5 x^{2} + 4 + C

Solution steps

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

Apply u-substitution

= \int \frac{1}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = x^{4} + 5 x^{2} + 4

= \frac{1}{2} ln x^{4} + 5 x^{2} + 4

Add a constant to the solution

= \frac{1}{2} ln x^{4} + 5 x^{2} + 4 + C

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