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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{1}{4 u} d u

= \int \frac{1}{4 u} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{4} ln ∣ u ∣

Substitute back

u = x^{4} + 5

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{1}{x ( 8 + ln ( x ) )})

x \to - 3 lim (\frac{x + 2}{x + 3})

x \to - 3 lim (\frac{x - 2}{x + 5})

\frac{d}{d x} (x^{\frac{1}{2}} y^{\frac{1}{2}})

x \to 8 lim (x^{3} - 6 x + 5)

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