What is the integral of (x^3)/(x^4+6) ?

The integral of (x^3)/(x^4+6) is 1/4 ln|x^4+6|+C

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integral of (x^3)/(x^4+6)

\int \frac{x ^{3}}{x ^{4} + 6} d x

\frac{1}{4} ln x^{4} + 6 + C

Solution steps

\int \frac{x ^{3}}{x ^{4} + 6} 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} + 6

= \frac{1}{4} ln x^{4} + 6

Add a constant to the solution

= \frac{1}{4} ln x^{4} + 6 + C

f (x) = ln (x^{7}), (1, 0)

\frac{d y}{d x} = \frac{2 x + 5 y}{x}

\frac{d}{d x} (\frac{6}{( x + 1 ) ^{4}})

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

What is the integral of (x^3)/(x^4+6) ?
The integral of (x^3)/(x^4+6) is 1/4 ln|x^4+6|+C