What is the integral of (8x^3)/(x^4+5) ?

The integral of (8x^3)/(x^4+5) is 2ln|x^4+5|+C

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integral of (8x^3)/(x^4+5)

\int \frac{8 x ^{3}}{x ^{4} + 5} d x

2 ln x^{4} + 5 + C

Solution steps

\int \frac{8 x ^{3}}{x ^{4} + 5} d x

Take the constant out:

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

= 8 \cdot \int \frac{x ^{3}}{x ^{4} + 5} d x

Apply u-substitution

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

= 8 \cdot \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

= 8 \cdot \frac{1}{4} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 8 \cdot \frac{1}{4} ln ∣ u ∣

Substitute back

u = x^{4} + 5

= 8 \cdot \frac{1}{4} ln x^{4} + 5

Simplify

8 \cdot \frac{1}{4} ln x^{4} + 5 : 2 ln x^{4} + 5

= 2 ln x^{4} + 5

Add a constant to the solution

= 2 ln x^{4} + 5 + C

x \to 0 - lim (- 1)

f (x) = 8 ln (x) - x^{2}

\frac{d}{d x} (x + 3 e^{- x})

\frac{\partial}{\partial y} (5 x^{\frac{y}{z}})

\int \frac{x ^{3} + 3 x + 4}{x ^{4} + 16 x ^{2}} d x

What is the integral of (8x^3)/(x^4+5) ?
The integral of (8x^3)/(x^4+5) is 2ln|x^4+5|+C