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

The integral of (x^3)/(x^8+5) is 1/(4sqrt(5))arctan((x^4)/(sqrt(5)))+C

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

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

\frac{1}{4 5} arctan (\frac{x ^{4}}{5}) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{4 ( u ^{2} + 5 )} d u

= \int \frac{1}{4 ( u ^{2} + 5 )} 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 ^{2} + 5} d u

Apply Integral Substitution

= \frac{1}{4} \cdot \int \frac{1}{5 ( v ^{2} + 1 )} d v

Take the constant out:

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

= \frac{1}{4} \cdot \frac{1}{5} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{1}{4} \cdot \frac{1}{5} arctan (v)

Substitute back

= \frac{1}{4} \cdot \frac{1}{5} arctan (\frac{x ^{4}}{5})

Simplify

\frac{1}{4} \cdot \frac{1}{5} arctan (\frac{x ^{4}}{5}) : \frac{1}{4 5} arctan (\frac{x ^{4}}{5})

= \frac{1}{4 5} arctan (\frac{x ^{4}}{5})

Add a constant to the solution

= \frac{1}{4 5} arctan (\frac{x ^{4}}{5}) + C

\frac{\partial}{\partial x} (e^{2 y} (3 e^{y} - 2 x))

\frac{s + 1}{( s + 6 ) ^{2} \cdot ( s + 2 )}

\frac{d}{d x} (- e^{x - 28} + 75)

\int - x - 2 d x

f (x) = 2 x e^{x} + (x^{2} - 14) \cdot e^{x}

What is the integral of (x^3)/(x^8+5) ?
The integral of (x^3)/(x^8+5) is 1/(4sqrt(5))arctan((x^4)/(sqrt(5)))+C